text:
\section{Introduction}
One of the key features of intelligence is the ability to combine and transfer
information between diverse domains and modalities \cite{crick1990towards, von1995binding, singer2001consciousness, treisman1996binding, seymour2009coding}.
In contrast, artificial intelligence research has made great progress in learning powerful
representations for \emph{individual} domains \cite{Han_2019,
oord2018representation, donahue2019large, oord2016wavenet, devlin2018bert,
brown2020language} that can even achieve superhuman performance on confined tasks
such as traffic sign recognition \cite{cirecsan2011committee,
cirecsan2012multi}, image classification \cite{he2016deep} or question
answering \cite{devlin2018bert}.
However, learning representations for different domains that also allow a
domain-to-domain transfer of information between them is significantly
more challenging \cite{Baltrusaitis_2019}: There is a trade-off between the
expressiveness of individual domain representations and their compatibility to
another to support transfer.  While for limited training data multimodal
learning has successfully trained representations for different domains
together \cite{srivastava2012multimodal,wang2015deep}, the overall most
powerful domain-specific representations typically result from training huge
models specifically for \emph{individual} challenging domains using massive
amounts of training data and computational resources, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot
\cite{donahue2019large,oord2016wavenet,brown2020language}.
\berttobigfirstpage
With the dawn of even more massive models like the recently introduced GPT-3
\cite{brown2020language}, where training on only a single domain already
demands most of the available resources, we must find new, creative ways to
make use of these powerful models, which none but the largest institutions
can afford to train and experiment with, and thereby utilize the huge amount of
resources and knowledge which are distilled into the model's representations---in other words,
we have to find ways to cope with "The Bitter Lesson" \cite{bitterlesson}.

Consequently, we seek a model for generic domain-to-domain transfer between
arbitrary fixed representations that come from highly complex, off-the-shelf,
state-of-the-art models
and we learn a domain translation that does not alter or retrain the individual
representations but retains the full capabilities of original expert models.
This stands in contrast to current influential domain transfer approaches
\cite{liu2019few,park2019semantic,zhang2020cross} that require learning or
finetuning existing domain representations to facilitate transfer between them.

Since different domains are typically not isomorphic to another, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot
translations between them are not uniquely determined, the domain translation
between fixed domain representations requires learning the corresponding
ambiguities. For example, there are many images which correspond to the same
textual description and vice versa.
To faithfully translate between domains, we employ a conditional invertible neural network (cINN) that also explicitly captures these transfer uncertainties. The INN conditionally learns a unique translation of one domain representation together with its complementary residual onto another. This generic network-to-network translation between arbitrary models can efficiently transfer between diverse state-of-the-art models such as transformer-based natural language model BERT
\cite{devlin2018bert} and a BigGAN \cite{brock2018large} for image synthesis to achieve competitive text-to-image translation, see Fig.~\ref{fig:berttobig}.

To summarize, our contributions are as follows: We (i) provide a generic
approach that allows to translate between fixed off-the-shelf model
representations, (ii) learns the inherent ambiguity of the domain translation,
which facilitates content creation and model diagnostics, and (iii) enables
compelling performance on various different domain transfer problems. We
(iv) make transfer between domains and datasets computationally affordable,
since our method does not require any gradient computations on the expert
models but can directly utilize existing representations.

\section{Related Work}
\label{sec:background}
The majority of approaches for deep-learning-based domain-to-domain translation
are based on generative models and therefore rely on Variational Autoencoders
(VAEs) \cite{VAE,VAE2}, Generative Adversarial Networks (GANs) \cite{gan},
autoregressive models \cite{ar}, or normalizing flows
\cite{papamakarios2019normalizing} obtained with invertible neural networks
(INNs) \cite{dinh2014nice,dinh2016density}. Generative models transform samples
from a simple base distribution, mainly a standard normal or a uniform
distribution, to a complex target distribution, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot the distribution of (a
subset of) natural images.

Sampling the base distribution then leads to the generation of novel content.
Recent works \cite{xiao2019generative,esser2020invertible} also utilize INNs
to transform the latent distribution of an autoencoder to the base
distribution.
A simple structure of the base distribution allows rudimentary control over the
generative process in the form of vector arithmetic applied to samples
\cite{radford2015unsupervised,NIPS2015_5845,shen2019interpreting,goetschalckx2019ganalyze},
but more generally, providing control over the generated content is formulated
as conditional image synthesis.
In its most basic form, conditional image synthesis is achieved by generative
models which, in addition to a sample from the base distribution, take class
labels \cite{mirza2014conditional,kingma2014semi} or attributes
\cite{he2019attgan} into account. More complex conditioning information are
considered in \cite{zhang2017stackgan,reed2016generative}, where textual
descriptions provide more fine-grained control over the generative process.  A
wide range of approaches can be characterized as image-to-image translations
where both the generated content and the conditioning information is given by
images. Examples for conditioning images include grayscale images
\cite{zhang2016colorful}, low resolution images \cite{ledig2017photo}, edge
images \cite{isola2017image}, segmentation maps
\cite{park2019semantic,chen2017photographic} or heatmaps of keypoints
\cite{Esser_2018,ma2017disentangled}.  Many of these approaches build upon
\cite{isola2017image}, which introduced a unified approach for image-to-image
translation. We take this unification one step further and provide an approach
for a wide range of conditional content creation, including class labels,
attributes, text and images as conditioning. In the case of image conditioning,
our approach can be trained either with aligned image pairs as in
\cite{isola2017image,chen2017photographic,park2019semantic} or with unaligned
image pairs as in
\cite{Zhu_2017,Lee_2018,choi2019stargan,Huang_2018,esser2019unsupervised}.
\modelneurips
While many works on generative models focus on relatively simple datasets
containing little variations, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot CelebA \cite{liu2015faceattributes}
containing only aligned images of faces, \cite{brock2018large,
donahue2019large} demonstrated the possibility to apply these models to
large-scale datasets such as ImageNet \cite{deng2009imagenet}. However, such
experiments require a computational effort which is typically far out of reach
for individuals. Moreover, the need to retrain large models for experimentation
hinders rapid prototyping of new ideas and thus slows down progress. Making use
of pre-trained neural networks can significantly reduce the computational
budget and training time. For discriminative tasks, the ability to effectively
reuse pre-trained neural networks has long been recognized
\cite{Razavian_2014,donahue2013decaf,yosinski2014transferable}. For generative
tasks, however, there are less works that aim to reuse pre-trained networks
efficiently. Features obtained from pre-trained classifier networks are used to
derive style and content losses for style transfer algorithms
\cite{Gatys_2016}, and they have been demonstrated to measure perceptual
similarity between images significantly better than pixelwise distances
\cite{mahendran2015understanding,zhang2018unreasonable}.
\cite{yosinski2015understanding,mahendran2016visualizing} find images which
maximally activate neurons of pre-trained networks and
\cite{santurkar2019image} shows that improved synthesis results are obtained
with adversarially robust classifiers. Instead of directly searching over
images, \cite{nguyen2016synthesizing} uses a pre-trained generator network of
\cite{dosovitskiy2016generating}, where it was used to reconstruct images from
feature representations.  However, these approaches are limited to neuron
activation problems, rely on per-example optimization problems, which makes
synthesis slow, and do not take into account the probabilistic nature of the
conditional synthesis task, where a single conditioning corresponds to multiple
outputs. To address this, \cite{nash2018inverting} learns an autoregressive
model conditioned on specific layers of pre-trained models and
\cite{shocher2020semantic} a GAN based decoder conditioned on a feature pyramid
of a pre-trained classifier. In contrast, our approach efficiently utilizes
pre-trained models, both for conditioning as well as for image-synthesis, such
that their combination provides new generative capabilities for content
creation through conditional sampling, without requiring the pre-trained models
to be aware of these emerging capabilities.

\section{Approach}
\label{sec:approach}

Our goal is to learn relationships and transfer between representations of
different domains obtained from off-the-shelf models, see
Fig.~\ref{fig:modelnips}.
To be generally applicable to complex state-of-the-art representations, we only
assume the availability of already trained models, but no practical access to
their training procedure due to their complexity \cite{devlin2018bert} or
missing components (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot a discriminator, which was not released for
\cite{donahue2019large}).
Let
$\mathcal{D}_{\x}$ and $\mathcal{D}_{\y}$ be two domains we want to transfer between.
Moreover, $f(x)$ denotes an expert model that has been trained to map
$x \in \mathcal{D}_{\x}$ onto desired outputs, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot class labels in case of
classification tasks, or synthesized images for generative image models. To
solve its task, a neural network $f$ has learned a latent
representation $\modelrep_{\oneenc} = \Phi(x)$ of domain $\mathcal{D}_{\x}$ in some
intermediate layer, so that subsequent layers $\Psi$ can then solve the task as
$f(x) = \Psi(\Phi(x))$. For $y \in \mathcal{D}_{\y}$ let
$g(y) = \Lambda(\Theta(y))$ be another, totally different model that
provides a feature representation vector $\modelrep_{\twoenc} = \Theta(y)$.

In general, we cannot expect a translation from $x$ to $y$ to be unique,
since two arbitrary domains and their representations are not necessarily
isomorphic.
For example, a textual description $x$ of an image $y$ usually leaves many
details open and the same holds in the opposite direction, since many
different textual descriptions are conceivable for the same image by focusing
on different aspects. This
implies a non-unique mapping from $\modelrep_{\oneenc}$ to $\modelrep_{\twoenc}$.
Moreover, much of the power of model $f$ trained for a specific task
stems from its ability to ignore task-irrelevant properties of $x$.
The invariances of $\modelrep_{\oneenc}$ with respect to $\modelrep_{\twoenc}$ further increase the
ambiguity of the domain translation. Obtaining a plausible $\modelrep_{\twoenc}$ for a given
$\modelrep_{\oneenc}$ is therefore best described probabilistically as sampling from
$p(\modelrep_{\twoenc} \vert \modelrep_{\oneenc})$.
Our goal is to model this process with a translation function $\tau$. Thus, we must
introduce a residual $v$,
such that for a given $\modelrep_{\oneenc}$, $v$ uniquely determines $\modelrep_{\twoenc}$
resulting in the
translation function $\tau$:
\begin{equation}
  \modelrep_{\twoenc} = \tau(v \vert \modelrep_{\oneenc})
\end{equation}

\newcommand{u}{u}
\paragraph{Learning a Domain Translation $\tau$:}
How can we estimate $v$? $v$ must capture all information of
$\modelrep_{\twoenc}$ not represented in $\modelrep_{\oneenc}$, but
no information that is already represented in $\modelrep_{\oneenc}$.
Hence, to infer $v$, we must take into account both
$\modelrep_{\twoenc}$, to extract information, and $\modelrep_{\oneenc}$, to discard information.
The unique determination of $\modelrep_{\twoenc}$ from $v$ for a given $\modelrep_{\oneenc}$ implies
the
existence of the inverse of $\tau$, when considered as a function of
$v$. Thus for every $\modelrep_{\oneenc}$, the inverse $\tau^{-1}(\cdot \vert
\modelrep_{\oneenc})$ of $\tau(\cdot \vert \modelrep_{\oneenc})$ exists,
\begin{equation}
  \label{eq:invertible}
  %
  v = \tau^{-1}(\modelrep_{\twoenc} \vert \modelrep_{\oneenc}) .
\end{equation}

This structure of $\tau$ is most naturally represented by a conditionally
invertible neural network (cINN), for which $\tau^{-1}$ can be explicitly
computed, and which we build from affine coupling \cite{dinh2016density},
actnorm \cite{kingma2018glow} and shuffling layers, see Sec.~\ref{subsec:cinnarch}. It then remains to derive a learning task which ensures that information of $\modelrep_{\oneenc}$ is discarded in
$v$. To formalize this goal, we consider training pairs
$\{(x, y)\} \subset \mathcal{D}_{\x} \times \mathcal{D}_{\y}$ and their corresponding features
$\{(\modelrep_{\oneenc}, \modelrep_{\twoenc})\}$ as samples from their joint distribution $p(\modelrep_{\oneenc},
\modelrep_{\twoenc})$. $v$ can then be considered as a random variable via the
process
\begin{equation}
  \label{eq:induced}
  v = \tau^{-1}(\modelrep_{\twoenc} \vert \modelrep_{\oneenc}), \quad \text{with }\modelrep_{\oneenc}, \modelrep_{\twoenc} \sim
  p(\modelrep_{\oneenc}, \modelrep_{\twoenc}).
\end{equation}
Then $v$ discards all information of $\modelrep_{\oneenc}$ if
$v$ and $\modelrep_{\oneenc}$ are independent. To achieve this independence, we
minimize the distance between the distribution $p(v \vert \modelrep_{\oneenc})$
induced by $\tau$ via Eq. ~\eqref{eq:induced} and some prior distribution
$q(v)$. The latter can be chosen arbitrarily as long as it is
independent of $\modelrep_{\oneenc}$, its density can be evaluated and samples can be
drawn. In practice we use a standard normal distribution.
Using the invertibility of $\tau$, we can then explicitly calculate the
Kullback-Leibler divergence between $p(v \vert \modelrep_{\oneenc})$ and
$q(v)$ averaged over $\modelrep_{\oneenc}$ (see Sec.~\ref{suppsec:objective} for the derivation):
\begin{align}
  \label{eq:loss}
  \mathbb{E}_{\modelrep_{\oneenc}} \operatorname{KL}(p(v \vert \modelrep_{\oneenc}) \vert q(v)) =
  &\mathbb{E}_{\modelrep_{\twoenc}, \modelrep_{\oneenc}}
  \left\{
    -\log q(\tau^{-1}(\modelrep_{\twoenc} \vert \modelrep_{\oneenc}))
    - \vert \det J_{\tau^{-1}}(\modelrep_{\twoenc} \vert \modelrep_{\oneenc}) \vert
  \right\}
  -H(\modelrep_{\twoenc} \vert \modelrep_{\oneenc}).
\end{align}
Here, $\det J_{\tau^{-1}}$ denotes the determinant of the Jacobian of $\tau^{-1}$ and $H$ is the (constant) data entropy. If $\tau$ minimizes Eq.~\eqref{eq:loss}, we have $p(v \vert
\modelrep_{\oneenc}) = q(v)$, such that the desired independence is achieved. Moreover, we
can now simply achieve the
original goal of sampling from $p(\modelrep_{\twoenc} \vert \modelrep_{\oneenc})$ by
translating from $\modelrep_{\oneenc}$ to $\modelrep_{\twoenc} = \tau(v \vert \modelrep_{\oneenc})$
with $v$ sampled from $q(v)$, which properly models
the inherent ambiguity. %

\paragraph{Interpretation as Information Bottleneck:}
One of the main goals of minimizing Eq.~\eqref{eq:loss} is the independence of
$v$ and $\modelrep_{\oneenc}$. While it is clear that this independence is achieved by a
minimizer, Eq.~\eqref{eq:loss} is also an upper bound on the mutual information
$I(v, \modelrep_{\oneenc})$ between $v$ and $\modelrep_{\oneenc}$. Thus, its
minimization works directly towards the goal of independence. Indeed, following
\cite{alex2016deep}, we have
\begin{equation}
  I(v, \modelrep_{\oneenc}) = \int_{v,\modelrep_{\oneenc}} p(v, \modelrep_{\oneenc}) \log \frac{p(v,
  \modelrep_{\oneenc})}{p(v) p(\modelrep_{\oneenc})} = \int_{v, \modelrep_{\oneenc}} p(v, \modelrep_{\oneenc}) \log
  p(v \vert \modelrep_{\oneenc}) - \int_{v} p(v) \log p(v)
\end{equation}
Positivity of the KL divergence implies $\int p(v) \log p(v)
\ge \int p(v) \log q(v)$, such that
\begin{equation}
  I(v, \modelrep_{\oneenc}) \le \int_{v, \modelrep_{\oneenc}} p(v, \modelrep_{\oneenc}) \log \frac{p(v
  \vert \modelrep_{\oneenc})}{q(v)} = \mathbb{E}_\modelrep_{\oneenc} \operatorname{KL}(p(v \vert
  \modelrep_{\oneenc}) \vert q(v)) = \text{Eq.~\eqref{eq:loss}}
  \label{eq:infobound}
\end{equation}
In contrast to the deep variational information bottleneck \cite{alex2016deep},
our use of a cINN has the advantage that it does not require the hyperparameter
$\beta$ to balance the independence of $v$ and $\modelrep_{\oneenc}$ against their
ability to reconstruct $\modelrep_{\twoenc}$. The cINN guarantees perfect reconstruction
abilities of $\modelrep_{\twoenc}$ due to its invertible architecture and it thus suffices
to minimize $I(v, \modelrep_{\oneenc})$ on its own.

\paragraph{Domain Transfer Between Fixed Models:}
At inference time, we obtain translated samples $\modelrep_{\twoenc}$ for given $\modelrep_{\oneenc}$
by sampling from the residual space $v$ given $\modelrep_{\oneenc}$ and then
applying $\tau$,
    \begin{equation}
      \modelrep_{\twoenc} \sim p(\modelrep_{\twoenc} \vert \modelrep_{\oneenc}) %
      \quad \iff \quad %
      v \sim q(v) %
      \label{eq:sample}
      ,\;
      \modelrep_{\twoenc}= \tau(v \vert \modelrep_{\oneenc}) .
    \end{equation}
After training our domain translator, transfer between $\mathcal{D}_{\x}$ and $\mathcal{D}_{\y}$ is thus
achieved by the following steps: (i) sample $x$ from $p(x)$, (ii) encode $x$
into the latent space $\modelrep_{\oneenc} = \Phi(x)$ of expert model $f$,
(iii) sample a residual $v$ from the prior $q(v)$, (iv)
conditionally transform $\modelrep_{\twoenc} = \tau(v \vert \modelrep_{\oneenc})$, and (v)
decode $\modelrep_{\twoenc}$ into the domain $\mathcal{D}_{\y}$ of the second expert model: $y =
\Lambda(\modelrep_{\twoenc})$.

Note that this approach has multiple advantages: (i) hidden representations
usually have lower dimensionality than $x$, which makes transfer between
arbitrary complex domains affordable, (ii) the cINN $\tau$ can
be trained by minimizing the negative log-likelihood, independent of the
domains $\mathcal{D}_{\x}$ and $\mathcal{D}_{\y}$, and (iii) the approach does not require
to take any gradients w.r.t\onedot} \def\dof{d.o.f\onedot the expert models $f$ and $g$.

\section{Experiments}
We investigate the wide applicability of our approach by performing experiments
with multiple domains, datasets and models: (1) text-to-image translation by
combination of BigGAN and BERT, (2) exploration of the reusability of a fixed
autoencoder combined with multiple models including a ResNet-50 classifier and a
DeepLabV2 \cite{chen2017deeplab} segmentation model for various image-to-image
translation tasks and diagnostic insights into the respective models, and (3)
comparison to existing methods for image modification and applications in
exemplar-guided and unpaired image translation tasks.
As our method does not require gradients w.r.t\onedot} \def\dof{d.o.f\onedot the models $f$
and $g$, training of the cINN can be conducted on a single Titan X GPU.

\berttobiginception
\subsection{Translation to BigGAN}
\label{subsec:transtobig}
This section is dedicated to the task of using a popular but computationally expensive to train expert model as an image generator: BigGAN
\cite{brock2018large}, achieving state-of-the-art FID scores \cite{heusel2017gans} on the ImageNet dataset. As most GAN frameworks in general and BigGAN in particular do not
include an explicit encoder into a latent space, we aim to provide an encoding from an arbitrary domain by using an appropriate expert model $f$.
Aiming at the reusability of a fixed BigGAN $g$, and given the hidden
representation $\modelrep_{\oneenc} = \Phi(x)$ of the expert model $f=\Psi
\circ \Phi$, we want to find a mapping between $\modelrep_{\oneenc}$ and the latent
space $\modelrep_{\twoenc}$ of BigGAN's generator $\Lambda$, where, in accordance with
Fig.~\ref{fig:modelnips}, $\Theta \equiv \mathbb{1}$ and $g = \Lambda$.
Technical details regarding the training of our cINN can be found in
Sec.~\ref{subsec:berttobigdetails}. \newline
\vspace{-0.65cm}
\paragraph{BERT-to-BigGAN Translation:}
The emergence of transformer-based networks \cite{vaswani2017attention} has led to an immense leap in the field of natural language processing, where a popular model is the so-called BERT model. Here, we make use of a variant of the original model, which modifies BERT such that it produces a latent space in which input sentences can be compared for similarity via the cosine-distance measure \cite{reimers-2019-sentence-bert}. We aim to combine this representational power with the synthesis capabilities of BigGAN and thus train our model $\tau$ to map from the language representations $\modelrep_{\oneenc} = \Phi(x)$ into the latent space $\modelrep_{\twoenc}$ of BigGAN's generator as described above; hence $f = \Phi$ and $\Psi = \mathbb{1}$. During training, access to textual descriptions is obtained by using a captioning model as in \cite{xu2015show}, trained on the COCO \cite{lin2014microsoft} dataset. In a nutshell, at training time, we sample $\modelrep_{\twoenc}$, produce a corresponding image $\Lambda(\modelrep_{\twoenc})$, utilize \cite{xu2015show} to produce a text-caption $x$ describing the image and subsequently produce a sentence representation $\modelrep_{\oneenc} = \Phi(x)$ which we use to minimize the overall objective Eq.~\eqref{eq:loss}.
Results can be found in Fig.~\ref{fig:berttobig} and
Tab.~\ref{tab:bertinception}. Our model captures both fine-grained and coarse
descriptions (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot blue bird vs. yellow bird; school bus vs. pizza) and is able
to synthesize images with highly different content, based on given textual
inputs $x$. Although not being trained on the COCO images,
Tab.~\ref{tab:bertinception} shows that our model is highly competitive and
on-par with the state-of-the art in terms of Inception
\cite{salimans2016improved} and FID \cite{heusel2017gans} scores where
available.

\subsection{Repurposing a single target generator for different
source domain models}
\label{subsec:reuse}
Here, we train the cINN $\tau$ conditioned on hidden representations of
networks such as classifiers and segmentation models, and thereby show that
standard classifiers on arbitrary source domains can drive the same generator
to create content by transfer.
Refering to
Fig.~\ref{fig:modelnips}, this means that $f$ is represented by a
classifier/segmentation model, whereas $\Lambda$ is a decoder of an
autoencoder that is pretrained on a dataset of interest. Furthermore, we
evaluate the ability of our approach to combine a single, powerful domain
expert (the autoencoder) with different source models to solve a variety of
image-to-image translation tasks.
\animaltable
The autoencoder is trained on a combination of all carnivorous animal classes
in ImageNet and images of the \emph{AwA2} dataset \cite{xian2018zero}, split
into 211306 training images and 10000 testing images, which we call the
\emph{Animals} dataset. The details regarding architecture and training of this
autoencoder are provided in Sec.~\ref{suppsec:aearch}.

\paragraph{Image-to-Image Translation:}
In Fig.~\ref{fig:animaltable}, we investigate the translation from different
source domain models $\Phi$ onto the same generator $\Lambda$
using our cINN $\tau$.
In Fig.~\ref{tab:segtoimgargmax}, $f$ is a segmentation network trained
on COCOStuff, and $\Phi = f$, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot $\modelrep_{\oneenc}$ is given
by the final segmentation output of the network. This case corresponds to a
translation from segmentation masks to images and we observe that our approach
can successfully fuse the segmentation model with the autoencoder to obtain a
wide variety of generated image samples corresponding to a given segmentation mask.
Fig.~\ref{tab:segtoimglogits} uses the same segmentation network for
$f$, but this time, $\Phi$ are the logit predictions of the
network (visualized by a projection to RGB values). The diversity of
generated samples is greatly reduced compared to Fig.~\ref{tab:segtoimgargmax},
which indicates that logits still contain a lot of information which are not
strictly required for segmentation, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot the color of animals.
This shows how different layers of an expert can be selected to obtain more
control over the synthesis process.

In Fig.~\ref{tab:sobeltoimg}, we consider the task of translating edge images
to natural images. Here, $x$ is obtained through the Sobel filter, and we
choose a ResNet pretrained for image classification on stylized ImageNet as a
domain expert for edge images, as it has shown sensitivity to shapes
\cite{geirhos2018imagenet}.
This combination of $\Phi$ and $\Lambda$ through $\tau$ enables
edge-to-image translation.
Fig.~\ref{tab:inpainting} shows an image inpainting task, where $x$ is a
masked image. In this case, large portions of the shape are missing from the
image but the unmasked regions contain texture patches. This makes a ResNet
pretrained for image classification on ImageNet a suitable domain expert due to
its texture bias. The samples demonstrate that textures are indeed faithfully
preserved.

Furthermore, we can employ the same approach for generative superresolution.
Fig.~\ref{fig:superres} shows the resulting transfer when using our method for combining two
autoencoders, which are trained on different scales. More precisely, $f$ is an autoencoder trained
on images of size $32 \times 32$, while $g$ is an autoencoder of $256 \times 256$ images.
The samples show that the model captures the ambiguities w.r.t\onedot} \def\dof{d.o.f\onedot this translation and thereby enables efficient superresolution.
\vspace{-0.15cm}
\superres
\paragraph{Model Diagnosis:}
\landscapeslayers
Besides being applicable for content creation, our approach to recovering the
invariances of $f$ can also serve as a diagnostic tool for model
interpretation.
By comparing the generated samples $y = \Lambda(\modelrep_{\twoenc})$ (see
Eq.~\eqref{eq:sample}) conditioned on representations $\modelrep_{\oneenc} = \Phi(x)$
extracted from \emph{different layers} of $f$, we see how the invariances increase
with increasing layer depth and can thereby visualize what the model has
learned to ignore. Using the same segmentation model $f$ and autoencoder $g$ as
described above, we visualize the invariances and model representations for
different layers on the \emph{Animals} and a web-scraped \emph{Landscapes}
dataset. For the latter, Fig.~\ref{fig:landscapeslayers} demonstrates how the
model $f$ has learned to discard information in later layers; \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot the
variance in the synthesized outputs increases (such as different lightings or
colors for the same scene). The corresponding experiment on the \emph{Animals}
dataset is presented in Sec.~\ref{suppsec:ablation}.
There, we also study the importance of faithfully modeling ambiguities of the translation
by replacing the cINN with an MLP,
which in contrast to the cINN fails to translate deep representations.

Note that all results in Fig.~\ref{fig:animaltable} and
Fig.~\ref{fig:landscapeslayers},~\ref{fig:animalswap} were obtained by
combining a single, generic autoencoder $g$, which has no capabilities
to process inputs of $\mathcal{D}_{\x}$ on its own, and different domain experts
$f$, which possess no generative capabilities at all. These results
demonstrate the feasibility of solving a wide-range of image-to-image tasks
through the fusion of pre-existing, task-agnostic experts on image domains
$\mathcal{D}_{\x}, \mathcal{D}_{\y}$.  Moreover, choosing different layers of the expert
$f$ provides additional, fine-grained control over the generation
process.

\subsection{Evaluating image modification capabilities of our generic approach}
\label{subsec:imgmod}
\celebtab
\paragraph{Attribute Modification:}
To compare our generic approach against task-specific approaches, we compare its ability for
attribute modification on face images to those of \cite{choi2018stargan}. We
train the same autoencoder $g$ as in the previous section on CelebA \cite{liu2015faceattributes}, and directly use
attribute vectors as $\modelrep_{\oneenc}$.  For an input image $y$ with
attributes $\modelrep_{\oneenc}$, we synthesize versions with modified
attributes $\modelrep_{\oneenc}^*$. In each column of
Fig.~\ref{fig:celebtab}, we flip the binary entry of the corresponding
attribute to obtain $\modelrep_{\oneenc}^*$. To obtain the modified image, we
first compute $\modelrep_{\twoenc} = \Theta(y)$ and use its corresponding
attribute vector $\modelrep_{\oneenc}$ to obtain its attribute invariant representation $v =
\tau^{-1}(\modelrep_{\twoenc} \vert \modelrep_{\oneenc})$. We then mix it
again with the modified attribute vector to obtain $\modelrep_{\twoenc}^* =
\tau(v \vert \modelrep_{\oneenc}^*)$, which can be readily
decoded to the modified image $y^* = \Lambda(\modelrep_{\twoenc}^*)$.

Qualitative results in Fig.~\ref{fig:celebtab} demonstrate successful
modification of attributes. In comparison to \cite{choi2018stargan}, our approach
produces more coherent changes, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot changing gender causes changes in hair
length and changes in the beard attribute have no effect on female faces.
This demonstrates the advantage of fusing attribute information on a
low-dimensional representation of a generic autoencoder.
Overall, our approach produces images of higher quality, as
demonstrated by the FID scores \cite{heusel2017gans} in Fig.~\ref{fig:celebtab}.
Note that FID-scores are calculated w.r.t\onedot} \def\dof{d.o.f\onedot the complete dataset, explaining the
high FID scores for attribute \emph{glasses}, where images consistently
possess a large black area.

\vspace{-0.4cm}
\paragraph{Exemplar-Guided Translation:}
\humananimalswap
Another common image modification task is exemplar-guided
image-to-image translation \cite{park2019semantic}, where the semantic content and
spatial location is determined via a label map and the style via an exemplar
image. To approach this task, we utilize the same segmentation model and
autoencoder as in Sec.~\ref{subsec:reuse}. As before, we use the last layer of
the segmentation model to represent semantic content and location. For a given
exemplar $y$, we can then extract its residual $v =
\tau^{-1}(\Theta(y) \vert \Phi(y))$ and the
segmentation representation $\modelrep_{\oneenc} = \Phi(x)$ of another image
$x$. Due to the independence of $v$ and $\modelrep_{\oneenc}$, we can now transfer
$\modelrep_{\oneenc}$ under the guidance of $v$ to a recombined $\modelrep_{\twoenc}^* =
\tau(v \vert \modelrep_{\oneenc})$ which is readily decoded to an image $y^* =
\Lambda(\modelrep_{\twoenc}^*)$ as shown in Fig.~\ref{fig:animalswap}.
\vspace{-0.25cm}
\paragraph{Unsupervised Disentangling of Shape and Appearance:}
Our approach also allows for an unsupervised variant of the previous task,
as shown in
Fig.~\ref{fig:unsupswap}. Here, we use a random spatial deformation $d$
(see Sec.~\ref{supp:unsupervised}) to define $\Phi=\Theta\circ d$. Thus,
$\modelrep_{\oneenc}$ and $\modelrep_{\twoenc}$ share the same appearance but differ in pose, which is
distilled into $v$. For exemplar
guided synthesis, the roles of $x$ and $y$ are now swapped.
\vspace{-0.25cm}
\paragraph{Unpaired Image Translation:}
Fig.~\ref{fig:secondpage} demonstrates results of our approach
applied to unpaired image-to-image translation. Here, we use the same setup as
for attribute modification, but train various cINN models for unpaired transfer between the following datasets: CelebA and AnimalFaces-HQ \cite{choi2019stargan}, FFHQ and CelebA-HQ \cite{karras2018progressive}, Anime \cite{danbooru2019Portraits} to CelebA-HQ/FFHQ and Oil Portraits to CelebA-HQ/FFHQ. Details regarding the training procedure can be found in the Sec.~\ref{supp:unpaired}.
\section{Conclusion}
\vspace{-0.25cm}
This paper has addressed the problem of generic domain transfer between
arbitrary fixed off-the-shelf domain models. We have proposed a conditionally
invertible domain translation network that faithfully transfers between
existing domain representations without altering them. Consequently, there is
no need for costly or potentially even infeasible retraining or finetuning of
existing domain representations. The approach is \emph{(i) flexible:} Our cINN
for translation as well as its optimization procedure are independent from the
individual domains and, thus, provide plug-and-play capabilities by allowing to
plug in arbitrary existing domain representations; \emph{(ii) powerful:}
Enabling the use of pretrained expert domain representations outsources the
domain specific learning task to these models. Our model can thus focus on the
translation alone which leads to improvements over previous approaches;
\emph{(iii) convenient and affordable}: Users can now utilize powerful,
pretrained models such as BERT and BigGAN for new tasks they were not designed
for, with just a single GPU instead of the vast multi-GPU resources required
for training such domain models. Future applications include transfer between
diverse domains such as speech, music or brain signals.

\creativityfigure
\FloatBarrier
\clearpage

\section*{Broader Impact}

\begin{description}
\item[Environmental and Economic Aspects:]\hfill\\ Single training runs of large scale
    models have a large environmental footprint due to the massive computational
    requirements. It is therefore unreasonable to repeat this effort for every
    new application. Instead we allow to reuse powerful models, leading to significant reductions in computational demands.
  \item[Boosting serendipity:]\hfill\\ New (scientific) knowledge arises where seemingly unrelated entities are brought together to study their relationship (cf. the 'double projection' proposed by Heinrich Wölfflin a century ago for art history and other image disciplines to easily contextualize diverse imagery of different cultures). Allowing to efficiently connect expert models for diverse data and problems, thus promises to provide the basis for new directions of future research.
  \item[Interdisciplinary research:]\hfill\\ One of the main stumbling blocks for interdisciplinary research (e.g. vision and language) is to bring together expert models for widely different domains. State-of-the-art models are typically developed in and for the individual disciplines. Being able to efficiently combine these disciplinary expert models to solve interdisciplinary problems promises to be an enabling factor for more effective cross-disciplinary research.
  \item[Social equality:]\hfill\\ Training state-of-the art models is typically so costly that only wealthy institutions can afford their transfer and application to different input domains or other subsequent research that would require to retrain them. Computationally efficient transfer of existing models with no need for costly retraining therefore increases the opportunities for economically weaker institutions and countries to have research programs in this field.
  %
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  \item[Increasing applicability and impact of research output:]\hfill\\ Large scale research is often funded by public resources and thus there
    is a responsibility to make results accessible to the public. While
    pretrained models are often shared publicly, the scope in which they can be
    applied is significantly widened by our approach.
  \item[Content creation and manipulation:]\hfill\\ Domain transfer applied to controlled image synthesis and modification has wide applicability in the creative industry and beyond. However, it can also be misused for forgery and manipulation.
\end{description}

\begin{ack}
  This work has been supported in part by the German Research Foundation (DFG)
  projects 371923335, 421703927, and EXC 2181/1 - 390900948, the German federal
  ministry BMWi within the project ``KI Absicherung'' and a hardware donation
  from NVIDIA Corporation.
\end{ack}

\FloatBarrier
\clearpage
{\small
\bibliographystyle{ieee_fullname}


text:
\section{Introduction}

Let $S_g$ denote a closed, orientable surface of genus $g \geq 2$ and $\Mod(S_g)$ its mapping class group.
The goal of this paper is to compare two numbers associated to a pseudo-Anosov element $f$ of $\Mod(S_g)$:
\begin{itemize}
 \item $\kappa(f)$, the dimension of the subspace of $H_1(S_g;\R)$ fixed by $f$, and
 \item $h(f)$, the entropy of $f$.
\end{itemize}
Note that $0 \leq \kappa(f) \leq 2g$ and that $h(f)$ equals the logarithm of the stretch factor (or dilatation) $\lambda(f)$ (see \cite{primer} for the basic definitions).  Also, $\kappa(f)+1$ is the first betti number of the mapping torus associated to $f$.

For every $(k,g)$ with $g \geq 2$ and $0 \leq k \leq 2g$, we set
\[ L(k,g) = \textrm{min} \{ h(f) \mid f \colon S_g \to S_g \mbox{ and } \kappa(f) \geq k \}.\]

For two real-valued functions $F$ and $G$, we write $F \asymp G$ if there is a universal constant $C$ so that $F/C \leq G \leq CF$.

\begin{maintheorem}
The function $L(k,g)$ satisfies $L(k,g) \asymp \frac{k+1}{g}$.
\end{maintheorem}

Explicit constants are given in Theorem~\ref{T:homology_bound} below. The seminal result about $L(k,g)$ is due to Penner \cite{Pe}, who proved that $L(0,g) \asymp 1/g$.  With Farb, the second and third authors proved $L(2g,g) \asymp 1$.  Ellenberg asked \cite{Ellenberg} whether the $L(k,g)$ interpolate between $L(0,g)$ and $L(2g,g)$ in the sense that $L(k,g) \geq C(k+1)/g$ for some constant $C$.  Our result in particular answers his question while also bounding $L(k,g)$ from above.

\p{Explicit constants} For $f \in \Mod(S_g)$ and any field $\F$, we can define $\kappa_\F(f)$ to be the dimension of the subspace of $H_1(S_g;\F)$ fixed by $f$ and we can define $L_\F(k,g)$ similarly to $L(k,g)$, with $\kappa$ replaced by $\kappa_\F$.  The following theorem gives explicit constants as demanded by our main theorem and also generalizes our main theorem to arbitrary field coefficients.

\begin{theorem}\label{T:homology_bound} Let $\F$ be any field.  For all $g \geq 2$ and $0 \leq k \leq 2g$ we have:
\[  .00031\left ( \frac{k+1}{2g-2} \right ) \leq L_\F(k,g) \leq \twoentropy \left ( \frac{k+1}{2g-2} \right ) .\]
\end{theorem}
We use the denominator $2g-2$ instead of $g$ here because in our proofs of both the upper and lower bounds, it will be natural to consider the normalized entropy of a pseudo-Anosov mapping class $f \in \Mod(S_g)$, namely, $|\chi(S_g)| h(f)$.

\p{Comparison with previously known constants}   The best known constants for the theorems about $L(0,g)$ and $L(2g,g)$ are as follows:
\begin{alignat*}{6}
\frac{\log(2)}{6} \left ( \frac{1}{2g-2} \right ) & \ \ \leq \ \  & L(0,g) \, \,  & \ \ \leq \ \ && \log \left( \varphi^4 \right) \left ( \frac{1}{2g-2} \right ) \\
.197 & \ \ \leq \ \ &  L(2g,g) & \ \ \leq \ \ && \log(62)
\end{alignat*}
where $\varphi = (1+\sqrt{5})/2$ is the golden ratio.
The constant $\log(2)/3 \approx .231$ is due to Penner \cite{Pe}; see also McMullen \cite{Mc}.  The constant $\log(\varphi^4) \approx 1.925$ comes from the work of Aaber--Dunfield \cite{AD}, Hironaka \cite{Hir}, and Kin--Takasawa \cite{KT} who independently constructed examples proving
\[ \limsup_{g \to \infty} \ g \cdot L(0,g) \leq \log \left(  \varphi^2 \right).\]
This statement does not immediately imply the upper bound above, but appealing to Hironaka's construction and the work of Thurston \cite{ThNorm}, Fried \cite{Fr}, and McMullen \cite{Mc} we can use elementary calculus to promote this asymptotic statement to the given bound for all $g$; see Proposition~\ref{P:limsup to sup} in the appendix.  The constants $.197$ and $\log(62) \approx 4.127$ for $L(2g,g)$ are due to Farb and the second and third authors \cite{FLM1}.

A priori, we have $L(0,g) \leq L(k,g) \leq L(2g,g)$ and so the previously known bounds automatically give:
\[ \frac{\log(2)}{6} \left ( \frac{1}{2g-2} \right ) \leq  L(k,g) \leq \log(62).\]
The upper bound in Theorem~\ref{T:homology_bound} improves on the upper bound of $\log(62)$ as soon as $g$ is slightly larger than $k$ (specifically, $g > (k+1) \log(64)/\log(62) +1$).  In order for our Theorem~\ref{T:homology_bound} to improve on Penner's lower bound, we need $k > 372$, hence $g > 186$.   On the other hand, the argument for the lower bound can be applied under fairly mild hypotheses to provide an improvement on Penner's lower bound; see Corollary~\ref{C:CSmod2}.  Of course, the point of Theorem~\ref{T:homology_bound} is the asymptotic behavior.

\p{Fixed subspaces of a fixed dimension.} In the definition of $L(k,g)$ one might be inclined to replace the inequality with an equality, that is, to consider the smallest entropy among pseudo-Anosov elements of $\Mod(S_g)$ fixing a subspace of $H_1(S_g;\R)$ whose dimension is exactly equal to $k$.  Our Main Theorem is still valid with this definition; we explain the necessary modifications to the proof at the end of Section~\ref{sec:count}.  In Penner's original work there are no constraints on the action of homology, and in the work of the last two authors with Farb we have the strictest possible constraint.  As such, our  $L(k,g)$ best interpolates between these two situations.

\p{Congruence subgroups} Let $\Mod(S_g)[m]$ denote the level $m$ congruence subgroup of $\Mod(S_g)$, that is, the kernel of the natural action of $\Mod(S_g)$ on $H_1(S_g;\Z/m\Z)$.  We write $L(\Mod(S_g)[m])$ for the minimum entropy over all pseudo-Anosov elements of $\Mod(S_g)[m]$.   In work with Farb, the second and third author showed for fixed $m \geq 3$ that $L(\Mod(S_g)[m]) \asymp 1$; \cite{FLM1}.  Setting $k=2g$ and $\F=\F_2$ (the field with $2$ elements), $L_{\F_2}(2g,g) = L(\Mod(S_g)[2])$, and so our main theorem extends this result to the case of $m=2$, cf.~\cite[Question 2.9]{FLM1}.

\medskip

\p{Counting conjugacy classes}  The following theorem can be viewed as a refinement of a theorem of the second and third authors \cite[Theorem 1.3]{LM}.  In the statement, let $\mathcal{G}_{g,k}(L)$ denote the number of conjugacy classes of pseudo-Anosov mapping classes $f \in \Mod(S_g)$ with $\kappa(f) = k$ and $h(f) < L \cdot (k+1)/(2g-2)$.

\begin{theorem} \label{T:homology_count}
Let $k \geq 0$ and let $L \geq \entropy$.  There are constants $c_1,c_2>0$ so that
\[ |\mathcal{G}_{g,k}(L)| \geq c_1g^k-c_2. \]
\end{theorem}

Conjugacy classes of pseudo-Anosov homeomorphisms in $\Mod(S_g)$ are in bijection with geodesics in the moduli space of Riemann surfaces of genus $g$ and the entropy of a pseudo-Anosov mapping class is equal to the length of the corresponding geodesic in the Teichm\"uller metric on moduli space.  There is a canonical surface bundle on moduli space whose monodromy is given by the correspondence between loops in moduli space and elements of $\Mod(S_g)$.    Theorem~\ref{T:homology_count} can thus be viewed as an estimate on the number of short geodesics in moduli space where the associated monodromy fixes a subspace of dimension $k$.

\p{Acknowledgments} We would like to thank Jeffrey Brock, Ken Bromberg, Sadayoshi Kojima, and Greg McShane for helpful conversations.

\section{Lower bounds}

The following proposition is the main goal of this section.  It implies the lower bound in Theorem~\ref{T:homology_bound} and in fact generalizes it to the case of an arbitrary surface of finite type (that is, a surface obtained from a compact surface by deleting finitely many points from the interior).

\begin{proposition} \label{P:homology_bound_gen}
Let $S$ be a surface of finite type and $\F$ a field.  If $f \in \Mod(S)$ is pseudo-Anosov, then
\[ .00031 \left(\frac{\kappa_\F(f) + 1}{|\chi(S)|}\right ) \leq h(f).\]
\end{proposition}

The proof requires a few preliminary facts.  For a hyperbolic 3-manifold $M$ with finite volume, we denote by $\vol(M)$ the hyperbolic volume.  Also, for any field $\F$ we denote by $b_1(M;\F)$ the first Betti number of $M$ with coefficients in $\F$.

\begin{proposition} \label{P:volume_b1}
Let $\F$ be any field.  If $M$ is a complete, orientable, hyperbolic $3$-manifold of finite volume, then
\[ b_1(M;\F) \leq 334.08 \cdot \vol(M). \]
\end{proposition}

Gelander proved  \cite[Corollary 1.3]{Gelander} there exists a constant $D > 0$ so that $b_1(M;\F) \leq D \cdot \vol(M)$ (see also \cite{BGLS}).  Gelander's result applies in much greater generality, but the proof there does not provide an explicit constant.

For a Riemannian manifold, the $\epsilon$-thick part $M_{\geq \epsilon}$ is the subset of $M$ with injectivity radius greater than or equal to $\epsilon/2$.  An $\epsilon > 0$ is a \emph{Margulis constant} for a hyperbolic manifold $M$ if
\begin{itemize}
\item $M_{\geq \epsilon}$ is the complement of a union of open solid tori and product neighborhoods of the ends of $M$, and
\item for every $x \in M_{\geq \epsilon}$, the open ball $B_{\epsilon/2}(x)$ in $M$ is path isometric to an open ball of radius $\epsilon/2$ in hyperbolic $3$--space.
\end{itemize}

To prove Proposition~\ref{P:volume_b1}, we will need a theorem of the first author with Culler and Shalen \cite[Corollary 4.2 and Lemma 5.2]{ACS}.

\begin{theorem}[Agol--Culler--Shalen]
\label{thm:acs}
Let $M$ be a complete hyperbolic manifold of finite volume.
\begin{enumerate}
\item If each 2-generator subgroup of $\pi_1(M)$ is either free or free abelian, then $\log(3)$ is a Margulis number for $M$.
\item If $M$ satisfies either of the conditions
\begin{itemize}
 \item $b_1(M;\Q) \geq 3$, or
 \item $M$ is closed and $b_1(M;\F_p) \geq 4$ for some prime $p$,
\end{itemize}
then each 2-generator subgroup of $\pi_1(M)$ is either free or free abelian; in particular, $\log(3)$ is Margulis constant for $M$.
\end{enumerate}
\end{theorem}

We will also need to extend this theorem to one additional case.

\begin{proposition}
\label{P:ACSlog3}
Let $p$ be any prime.  If $M$ is a complete, noncompact hyperbolic manifold of finite volume with $b_1(M;\F_p) \geq 5$, then $log(3)$ is a Margulis number for $M$.
\end{proposition}

\begin{proof}

By the structure of the thick-thin decomposition of $M$ \cite[Corollary 5.2.10]{Th}, $M$ is homeomorphic to the interior of a compact manifold $M'$ with boundary $\partial M'$ homeomorphic to a union of $d \geq 1$ tori.  From Poincar\'e--Lefschetz duality and the long-exact sequence of the homology of the pair $(M',\partial M')$ it follows that the image of $H_1(\partial M';\F) \to H_1(M';\F) \cong H_1(M;\F)$ has dimension exactly $d$ for any field $\F$.   Thus, if $d \geq 3$ then $b_1(M;\Q) \geq 3$, and we may appeal to Theorem~\ref{thm:acs}(2).

It remains to consider the cases of $d \in \{1,2\}$.  Let $G$ be any 2-generator subgroup of $\pi_1(M)$.  The image of $H_1(G;\F_p)$ in $H_1(M;\F_p)$ is a subspace of dimension at most $2$, which together with the image of $H_1(\partial M';\F_p)$ spans a subspace of dimension at most $2+d$ in $H_1(M;\F_p)$.  Therefore, we can find a homomorphism from $\pi_1(M)$ onto $\F_p^r$, where $r \geq b_1(M;\F_p)-(2+d) \geq 3-d$ whose kernel contains both $G$ and $\pi_1(\partial M'_0)$ for each component $\partial M'_0 \subset \partial M'$.  Then the cover $\widetilde M' \to M'$ corresponding to the kernel has $dp^r \geq kp^{3-d} \geq 4$ boundary components and has fundamental group containing an isomorphic copy of $G$.  Appealing again to Theorem~\ref{thm:acs} completes the proof.
\end{proof}

\begin{proof}[Proof of Proposition \ref{P:volume_b1}]

Let $\F$ be any field.  To prove the proposition, we may assume that $b_1(M;\F) \geq 5$; indeed, the volume of any orientable hyperbolic 3-manifold is at least that of the Weeks manifold, which is approximately $.94$ \cite{GMM} and so $334.08 \cdot \vol(M)$ is always (much) greater than five.  When $b_1(M;\F) \geq 5$, then $b_1(M;\F_p) \geq 5$ for some prime $p$ (indeed, a matrix representing the action of $f$ on  $H_1(S_g;\F)$ has coefficients in the prime subfield of $\F$, and thus $\kappa_\F(f)$ depends only on the characteristic of $\F$).  Thus by Proposition \ref{P:ACSlog3}, $\epsilon = \log(3)$ is a Margulis number for $M$.

Since $\pi_1(M_{\geq \epsilon}) \to \pi_1(M)$ is surjective it suffices to show that $b_1(M_{\geq \epsilon};\F) \leq 334.08 \cdot \vol(M)$.  To this end, let $V = \{v_1, \dots, v_m \}$ be a maximal collection of points in $M_{\geq \epsilon}$ so that the distance in $M$ between any two points of $V$ is at least $\epsilon$ (that is, an $\epsilon$-net).   By maximality, $M_{\geq \epsilon}$ is contained in the $\epsilon$-neighborhood of $V$ in $M$.  Let $X$ be the Voronoi cell decomposition of $M$ defined by $V$ and $\Gamma$ the dual graph (i.e. the Delaunay graph).  Let $\Gamma_0 \subset \Gamma$ be the subgraph consisting of all edges of length at most $2 \epsilon$.  To complete the proof, we will show two things:
\begin{enumerate}
 \item the map $\pi_1(\Gamma_0,v_0) \to \pi_1(M_{\geq \epsilon},v_0)$ is surjective, and
 \item $b_1(\Gamma_0;\F) \leq 334.08 \cdot \vol(M)$.
\end{enumerate}
To prove the first statement, let $\gamma$ be any loop in $M_{\geq \epsilon}$ based at $v_0$.  We must find a loop in $\Gamma_0$ that is homotopic to $\gamma$ as a based loop.  Modifying $\gamma$ by homotopy in $M_{\geq \epsilon}$, we can assume $\gamma$ is transverse to the 2-skeleton of $X$; in particular, the intersection of $\gamma$ with any 2-cell lies in the interior of the 2-cell and there are finitely many such intersections.   Since $\gamma$ is contained in $M_{\geq \epsilon}$, it is always within distance $\epsilon$ of some $v_i$.  Therefore, any $2$--cell of $X$ that $\gamma$ crosses is dual to an edge with length at most $2\epsilon$.  It follows that we can homotope $\gamma$ (rel $v_0$) to the loop $\gamma' \subset \Gamma_0$ that traverses the edges determined by the $2$-cells $\gamma$ meets.

Now we bound $b_1(\Gamma_0;\F)$.  The number of vertices of $\Gamma_0$ is $m$. By definition of the $\{v_i\}$ and the fact that $\epsilon$ is a Margulis number for $M$ the $\epsilon/2$-balls $\{ B_{\epsilon/2}(v_j)\}_{j=1}^m$ are embedded and pairwise disjoint.  We thus have
\[ m \vol\left(B_{\epsilon/2}\right)  \leq \vol(M).\]
And using the formula $\vol(B_r) = \pi(\sinh 2r-2r)$ we calculate
\[ m < \vol(M)/0.234721.\]

The valence of any vertex of $\Gamma_0$ is bounded by the maximum number of points in a ball of radius $2 \epsilon$ with distance to the center at least $\epsilon$ and pairwise distance at least $\epsilon$.  This is the same as the maximum number of pairwise disjoint balls of radius $\epsilon/2$ that we can fit in the shell outside a ball of radius $\epsilon/2$ and inside a ball of radius $5 \epsilon/2$ with the same center.
This is bounded by the volume ratio
\[ \mathcal V = \frac{\vol(B_{5 \epsilon/2}) - \vol(B_{\epsilon/2})}{\vol(B_{\epsilon/2})} < 493.2244575.\]
The valence of $\Gamma_0$ is thus at most $\mathcal V$, and so $\Gamma_0$ has at most $\frac{\mathcal V}2 m$ edges.

The first betti number of a connected graph is given by $1 - (\# \text{vertices}) + (\# \text{edges})$.   Thus by the previous two paragraphs,
\[ b_1(\Gamma_0;\F) \leq 1+ \left(\frac{\mathcal V-2}{2}\right)m < 1 + \frac{491.2244575}{2\pi (0.234721)}\vol(M) < 1 + 333.08\vol(M). \]
Using the fact that a complete, hyperbolic manifold $M$ with $b_1(M;\F) \geq 5$ has volume at least 1 (see \cite{ACS,CM,PrzB1}), the proposition follows.
\end{proof}

\p{Remark} In the previous proof, we used a Voronoi decomposition instead of a good open cover because the latter would require an $\epsilon/2$-net instead of an $\epsilon$-net; this would result in a larger constant.

\bigskip

Proposition~\ref{P:homology_bound_gen} will follow easily from Proposition~\ref{P:volume_b1} and the following recent theorem of Kojima--McShane \cite{KM}; see also Brock-Bromberg \cite{BBWP}.   Kojima and McShane's theorem refines an earlier result of Brock \cite{BrockWP}, and builds on work of Schlenker \cite{S}.  To state it, we require a theorem of Thurston \cite{Tharxiv} which states that if $S$ is a hyperbolic surface of finite type and $f \in \Mod(S)$ is pseudo-Anosov, then the mapping torus $M = M_f$ admits a complete hyperbolic metric of finite volume; in particular, $\vol(M_f)$ makes sense and is finite.

\begin{theorem}  [Kojima--McShane] \label{T:Koj-McS}
Let $S$ be a surface of finite type.  If $f \in \Mod(S)$ is pseudo-Anosov, then
\[ \vol(M_f) \leq 3 \pi \,|\chi(S)|\, h(f).\]
\end{theorem}

\begin{proof}[Proof of Proposition~\ref{P:homology_bound_gen}]

It is an easy application of the Mayer--Vietoris long exact sequence that $b_1(M;\F) = \kappa_\F(f) + 1$ (decompose $S^1$ into the union of two intervals and pull this decomposition back to $M_f$).  Applying this fact, Proposition~\ref{P:volume_b1}, and Theorem~\ref{T:Koj-McS} in succession,  we have
\[ \kappa_\F(f)+1 = b_1(M;\F) \leq 334.08 \cdot \vol(M) \leq 334.08\cdot 3\pi\,|\chi(S)| \, h(f).\]
Since $1/(3 \pi \cdot  334.08) \approx .000317$, the proposition follows.
\end{proof}

As suggested to us by Peter Shalen, we can combine Theorem~\ref{T:Koj-McS} with the following result of Culler--Shalen to obtain an improvement on Penner's lower bound under a mild assumption on $f$.

\begin{theorem}[Culler--Shalen]
If $M$ is a hyperbolic 3-manifold with $b_1(M;\F_2) \geq 6$ then $\vol(M) \geq 3.08$.
\end{theorem}

\begin{corollary} \label{C:CSmod2}
Let $S$ be any surface of finite type and let $f \in \Mod(S)$.  If $\kappa_{\F_2}(f) \geq 5$, then $h(f) \geq  .326/|\chi(S)|$.
\end{corollary}

\section{Upper bounds} \label{S:examples}

The next proposition gives the upper bound in Theorem~\ref{T:homology_bound}.  Combining this proposition with Proposition~\ref{P:homology_bound_gen} immediately gives Theorem~\ref{T:homology_bound}.

\begin{proposition} \label{P:homology_bound_gen 2}
Let $\F$ be a field, let $g \geq 2$, and let $0 \leq k \leq 2g$.  There exists a pseudo-Anosov $f_{g,k} \in \Mod(S_g)$ with $\kappa_\F(f_{g,k}) \geq k$ and
\[h(f_{g,k})  \leq \twoentropy \left(\frac{k + 1}{2g-2}\right ) .\]
\end{proposition}

\p{Generating examples} We start by defining a collection of mapping classes
\[ \{f_g \in \Mod(S_g) \mid g \geq 2\} \cup \{f_2' \in \Mod(S_2)\} \]
from which all of our examples $f_{g,k}$ required for Proposition~\ref{P:homology_bound_gen 2} will be generated.  More precisely, each $f_{g,k}$ will be realized as the monodromy from some fibering of some mapping torus $M_{f_g}$ with $g = \lceil k/2 \rceil$ or of the mapping torus $M_{f_2'}$.

For each $g \geq 3$, we will describe explicit multicurves $A_g$ and $B_g$ in $S_g$, and then $f_g$ will be the difference of the two multitwists:
\[ f_g = T_{A_g} T_{B_g}^{-1}. \]
All of the $A_g,B_g$ pairs for $g \geq 3$ will be modeled on $A_3,B_3$ in $S_3$ shown in the left-hand drawing here:

\vspace*{2ex}

\begin{center}
\labellist
\small\hair 2pt
 \pinlabel {$\alpha_\ell$} [ ] at 114 48
 \pinlabel {$\beta_\ell$} [ ] at 113 11
 \pinlabel {$\alpha_m$} [ ] at 168 57
 \pinlabel {$\beta_m$} [ ] at 168 -1
 \pinlabel {$\alpha_r$} [ ] at 227 45
 \pinlabel {$\beta_r$} [ ] at 227 11
 \pinlabel {$\alpha$} [ ] at 71 -2
 \pinlabel {$\beta$} [ ] at 25 61
 \pinlabel {$\alpha$} [ ] at 189 -2
 \pinlabel {$\beta$} [ ] at 143 61
 \pinlabel {$\alpha$} [ ] at 241 46
 \pinlabel {$\beta$} [ ] at 241 11
 \pinlabel {$\alpha$} [ ] at 266 0
 \pinlabel {$\beta$} [ ] at 266 58
 \endlabellist
\noindent \includegraphics[scale=1.2]{pieces}
\end{center}

\vspace*{2ex}

We will build most of the other examples for $g \geq 3$ by gluing copies of the marked surfaces shown in the middle and right-hand pictures above (by a marked surface we mean a surface with labeled curves and arcs in it).

Let $P$ denote the marked surface in the middle and $Q$ the marked surface on the right.  We will need several versions of $P$; for any subset $J$ of $\{\ell,m,r\}$ let $P(J)$ denote the marked surface obtained from $P$ by excluding all curves and arcs whose labels have subscripts not in $J$; we will write, for instance, $P(\ell,r)$ instead of $P(\{\ell,r\})$.  The two pairs of curves labeled simply by $\alpha$ and $\beta$ will appear in each copy of $P$.

For $X_1,\dots, X_n \in \{P(J) \mid J \subseteq \{\ell,m,r\} \} \cup \{Q\}$, we will write $X_1 + \cdots + X_n$ for the marked surface obtained by gluing the $X_i$ end to end and capping the boundary of the resulting surface with two disks (this operation is not commutative!).  We will also write $c \cdot X_i$ for the sum of $c$ copies of $X_i$.  Because the six arcs in $P$ and $Q$ are consistently labelled, such a sum always results in a marked surface where each curve can be consistently labeled as an $\alpha$-curve or a $\beta$-curve, as long as each $P(J)$-piece with $r \in J$ is followed by either a $Q$-piece or a $P(J)$-piece with $\ell \in J$ (and similarly for the $P(J)$ with $\ell \in J$).
For example, $P(r) + Q + P(\ell)$ is the following marked surface of genus seven:

\begin{center}
\labellist
\small\hair 2pt
\endlabellist
\noindent \includegraphics[scale=1.2]{seven}
\end{center}

With this notation in hand, we can describe all of our $A_g$ and $B_g$ with $g \geq 3$ and $g \notin \{4,5,8\}$ as follows.  For $g=3$ we take $P(m)$ as above.  Then for any $k \geq 2$, we take
\[
\begin{cases}
P(m,r) + (k-2) \cdot P(\ell,r) + P(\ell) & g = 3k \\
P(r) + Q + (k-2) \cdot P(\ell,r) + P(\ell) & g = 3k+1 \\
P(r) + Q  + (k-2) \cdot P(\ell,r) + Q+ P(\ell) & g = 3k+5
\end{cases}
\]
We define $A_g$ and $B_g$ as the unions of the $\alpha$-curves and the $\beta$-curves, respectively.

The cases of $g \in \{4,5,8\}$ are described by the following pictures:

\vspace*{2ex}

\begin{center}
\labellist
\small\hair 2pt
\endlabellist
\noindent \includegraphics[scale=1.2]{exceptions}
\end{center}

\vspace*{2ex}

Here we did not label the curves as $\alpha$-curves or $\beta$-curves, but assigning the label of $\alpha$ or $\beta$ to any one of the curves determines the labels on the other curves (and it does not matter which of the two labelings we use).

Finally for $g=2$, we have $A_2,B_2$ and $A_2',B_2'$ shown in the left- and right-hand sides of the following figure, respectively:

\vspace*{4ex}

\labellist
\small\hair 2pt
 \pinlabel {$\beta$} [ ] at 50 55
 \pinlabel {$\alpha$} [ ] at 23 17
 \pinlabel {$\alpha$} [ ] at 180 55
 \pinlabel {$\beta_1$} [ ] at 142 14
 \pinlabel {$\beta_0$} [ ] at 169 34
\endlabellist
\centerline{\includegraphics[scale=1.1]{breadstick4}}

\vspace*{2ex}

\noindent and we define $f_2 = T_{A_2}T_{B_2}$ and $f_2' = T_{A_2'}T_{B_2'}$ (all twists are positive here).

\bigskip

\p{Properties of the generating examples} Our next goal is to bound from above the entropies of the $f_g$.  We will require the following theorem of Thurston \cite[Theorem 7]{ThMCG}.  For the statement we say that a pair of (minimally-intersecting) multicurves fills a closed surface if it cuts the surface into a disjoint union of disks.

\begin{theorem}[Thurston]
\label{theorem:thurston construction}
If $A=\{\alpha_1,\dots,\alpha_m\}$ and $B=\{\beta_1,\dots,\beta_n\}$ are multicurves that fill a surface $S$, then there is a homomorphism $\langle T_A , T_B \rangle \to \mathrm{PSL}(2,\R)$ given by
\[ T_A \mapsto \left( \begin{array}{cc} 1 & \sqrt{\mu} \\ 0 & 1\\ \end{array} \right) \quad \text{and}\quad T_B \mapsto \left( \begin{array}{cc} 1 & 0\\ -\sqrt{\mu} & 1\\ \end{array} \right), \]
where $\mu$ is the Perron--Frobenius eigenvalue of $NN^T$ and $N$ is the matrix $N_{ij} =  i(\alpha_i,\beta_j)$.  Moreover, an element of $\langle T_A , T_B \rangle$ is pseudo-Anosov if and only if its image is hyperbolic and in that case the entropy equals the logarithm of the spectral radius of the image.
\end{theorem}

\begin{lemma}
\label{lemma:dil}
Let $g \geq 2$.  Then $f_g$ is pseudo-Anosov with $h(f_g) < \entropy$.  Also, $f_2'$ is pseudo-Anosov with $h(f_2') < 2$.
\end{lemma}

\begin{proof}

We first deal with the case $g \geq 3$.  Under the map $\langle T_{A_g} , T_{B_g} \rangle \to \mathrm{PSL}(2,\R)$ from Theorem~\ref{theorem:thurston construction}, the image of $f_g$ is
\[  \left( \begin{array}{cc} 1 & \sqrt{\mu_g} \\ 0 & 1\\ \end{array} \right) \left( \begin{array}{cc} 1 & 0\\ -\sqrt{\mu_g} & 1\\ \end{array} \right)^{-1} \]
where $\mu_g$ is the Perron--Frobenius eigenvalue of the matrix $NN^T$ for $A_g$ and $B_g$.
The trace of this matrix is $2+\mu_g$.  Thus by Theorem~\ref{theorem:thurston construction} the entropy $h(f_g)$ is bounded above by $\log(\mu_g+2)$.  Therefore, to prove the lemma it is enough to show that $\mu_g \leq 62$ for all $g\geq 3$.

For the special cases $g \in \{3,4,5,8\}$ we can use a computer to check that $\mu_g < 62$; the largest is $\mu_8 \approx 61.978$.

The remaining values of $g \geq 3$ are covered by our general construction.  There are three cases, according to the residue modulo three, and we treat each in turn.  In these cases we will bound $\mu_g$ from above using the fact that the Perron--Frobenius eigenvalue of a Perron--Frobenius matrix is bounded from above by the maximum row sum of the matrix.

To determine the row sums of $NN^T$, we draw the labeled bipartite graph associated to $A_g$ and $B_g$.  This graph has one vertex for each connected component of $A_g$, one vertex for each connected component of $B_g$, and an edge between two vertices corresponding to simple closed curves with nonzero intersection; the labels are the geometric intersection numbers.  Here is the picture for the case of $g=3k$ (the labels here are suppressed since they are all equal to two):

\begin{center}
\noindent \includegraphics[scale=.4]{0mod3graph}
\end{center}

In general, to compute the row sum for $NN^T$ corresponding to a particular $\alpha$-curve we take the corresponding vertex of the bipartite graph and for each (possibly-backtracking) path of length two starting from that vertex we take the product of the two labels; the sum over all paths of length two is the row sum.  In the case where all of the graphs have the same label $n$, this number is equal to $n^2$ times the sum of the degrees of the vertices adjacent to the given vertex.

In the case of the graph given above for $g = 3k$ the degree of every vertex is bounded above by 4, and moreover, every vertex of degree $4$ is only adjacent to vertices of degree at most $3$.  Since the labels are all equal to 2, each row sum of $NN^T$ is at most $2^2 \cdot 3 \cdot 4 = 48<62$, as desired.

Next we consider the case where $g = 3k+2$ and $k \geq 3$.  In this case, the graph takes the following form (again all labels are equal to 2):

\begin{center}
\noindent \includegraphics[scale=.4]{2mod3graph}
\end{center}

In this case, there are vertices in the graph with degree 5, but the vertices adjacent to those have degrees equal to 1, 2, 2, 3, and 3, and so the row sums corresponding to the vertices of degree 5 are at most $2^2 \cdot (1+2+2+3+3)=44$.  The vertices of valence $4$ are only adjacent to vertices of valence at most $3$, so the row sums corresponding to vertices of valence 4 are again at most $48$ as in the previous case.   Finally, vertices of degree at most $3$ are adjacent to at most two vertices of degree at most 5 and a vertex of degree at most 3, and so the row sum associated to any remaining vertex is at most $2^2 \cdot (5+5+3) = 52$.  Thus all row sums are again less than 62.

For the case where $g = 3k+1$, the corresponding graph is a subgraph of the one for $g' = 3k'+2$ for any $k' \geq k$, and so the row sums are bounded above by the maximum row sums in that case, and are thus less than 62, as desired.

It remains to treat the cases of $f_2$ and $f_2'$.  In those cases the image in $\mathrm{PSL}(2,\R)$ is
\[  \left( \begin{array}{cc} 1 & \sqrt{\mu} \\ 0 & 1\\ \end{array} \right) \left( \begin{array}{cc} 1 & 0\\ -\sqrt{\mu} & 1\\ \end{array} \right) \]
and so the entropies are bounded above by $\log(\mu_2-2)$ and $\log(\mu_2'-2)$, where $\mu_2$ and $\mu_2'$ are the associated Perron--Frobenius eigenvalues.  For $f_2$ we have $NN^T = (8)(8)=(64)$ and so $\mu_2 = 64$; hence $h(f_2) < \log(62)$.  Similarly for $f_2'$ we have $NN^T = (2 \,\, 2)(2 \,\, 2)^T = (8)$, implying $\mu_2' = 8$ and hence $h(f_2') < \log(6) < 2$.
\end{proof}

For the next lemma, recall that for any $\vec v \in H_1(S_g)$ there is a transvection $\tau_{\vec v} : H_1(S_g) \to H_1(S_g)$ defined by
\[ \tau_{\vec v}(\vec w) = \vec w + \hat\imath(\vec w,\vec v) \vec v,\]
where $\hat\imath$ is the algebraic intersection number.   Note that $\tau_{\vec v} = \tau_{-\vec v}$.

The action on $H_1(S_g)$ of a Dehn twist $T_\alpha$ is a transvection $\tau_{[\vec \alpha]}$, where $\vec \alpha$ is $\alpha$ with some choice of orientation and $[\vec \alpha]$ is the associated homology class (see \cite[Proposition 6.3]{primer}); since $\tau_{\vec v} = \tau_{-\vec v}$ this transvection is well defined.

\begin{lemma}
\label{lemma:kappa}
For $g \geq 2$ we have $\kappa(f_g) = 2g$.  Also, $\kappa(f_2') = 2$.
\end{lemma}

\begin{proof}

First we treat the case of $f_g$ with $g \notin \{2,4,5,8\}$.  In these cases, $f_g$ has the form:
\[ \left(T_{\alpha_1} \cdots T_{\alpha_n}\right)\left(T_{\beta_1} \cdots T_{\beta_n}\right)^{-1}\]
where $[\vec \alpha_k] = \pm [\vec \beta_k]$ for all $k$ (after choosing arbitrary orientations).  Thus the action of $f_g$ on $H_1(S_g)$ is given by
\[ \left(\tau_{[\vec \alpha_1]} \cdots \tau_{[\vec \alpha_n]}\right)\left(\tau_{[\vec \alpha_1]} \cdots \tau_{[\vec \alpha_n]}\right)^{-1}.\]
Thus the action of $f_g$ on on $H_1(S_g)$ is trivial and so $\kappa(f_g) = 2g$.

The cases of $g \in \{4,5,8\}$ are similar, except in each case there is one additional Dehn twist about a separating simple closed curve.  Since a separating curve is null-homologous, the associated transvection is trivial and it follows again that $\kappa(f_g) = 2g$.

The cases of $f_2$ and $f_2'$ are also similar.  First $f_2$ is a product of Dehn twists about separating curves, so it acts trivially on $H_1(S_2)$.  For the case of $f_2'$ the 2-dimensional subspace of $H_1(S_2)$ that is preserved is the one spanned by the two nonseparating curves in $A_2' \cup B_2'$.
\end{proof}

\p{Mapping tori and surfaces of genus two in the fibered cone} As mentioned, our examples $f_{g,k}$ will all be derived from the mapping classes $f_g$ already defined.  The main vehicle for doing this is the following theorem of Thurston.

\begin{theorem}[Thurston] \label{T:thurston}
Let $M$ be a hyperbolic 3-manifold of finite volume.  There is a norm $\|\cdot\|$ on $H_2(M)$ with the following properties:
\begin{enumerate}
\item there is a set of maximal open cones $\C_1,\dots,\C_n$ in $H_2(M)$ and a bijection between the set of homotopy classes of connected fibers of fibrations $M \to S^1$ and the set of primitive integral points in the union of the $\C_i$;
\item the restriction of $\| \cdot \|$ to any $\C_i$ is linear;
\item if $\Sigma$ is a fiber in some fibration $M \to S^1$, then $\|[\Sigma]\| = -\chi(\Sigma)$; and
\item if a surface $\Sigma \subseteq M$ is transverse to the suspension flow associated to some fibration $M \to S^1$, then $[\Sigma]$ lies in the closure $\bar \C_i$ of the corresponding open cone $\C_i$.
\end{enumerate}
\end{theorem}

The norm $\|\cdot\|$ in Theorem~\ref{T:thurston} is now called the Thurston norm and the open cones $\C_i$ are called the \emph{fibered cones} of $M$.

Our next immediate goal is to use Theorem~\ref{T:thurston} to find surfaces of genus two that represent points in the closure of the fibered cones corresponding to our $f_g$.  We need a preliminary lemma.

\begin{lemma}
\label{lemma:boundary prelim}
Let $A = \{\alpha_1,\dots,\alpha_m\}$ and $B=\{\beta_1,\dots,\beta_n\}$ be multicurves in $S_g$.  Suppose that $\alpha_1$ and $\beta_1$ are homologous, disjoint simple closed curves that bound a subsurface of genus one, and that $\delta$ is a simple closed curve in $S_g$ with $i(\delta,\alpha_1) = i(\delta,\beta_1) = 1$ and all other intersection numbers $i(\delta,\alpha_k)$ and $i(\delta,\beta_k)$ equal to zero.  Then there are representatives of $\delta$ and $T_AT_B^{-1}(\delta)$ that are disjoint and bound a subsurface of genus one.
\end{lemma}

\begin{proof}

We first use the fact that the $T_{\alpha_i}$ pairwise commute and that the $T_{\beta_i}$ pairwise commute to write
\[ T_AT_B^{-1} = \left(T_{\alpha_2} \cdots T_{\alpha_m}\right)\left(T_{\alpha_1}T_{\beta_1}^{-1}\right)\left(T_{\beta_2}\cdots T_{\beta_n}\right)^{-1}.\]
Since the $\beta_i$ with $i > 1$ are disjoint from $\delta$, it follows that
\[ T_AT_B^{-1}(\delta) = \left(T_{\alpha_2} \cdots T_{\alpha_m}\right)\left(T_{\alpha_1}T_{\beta_1}^{-1}\right)(\delta).\]
It is straightforward to check under our assumptions that $T_{\alpha_1}T_{\beta_1}^{-1}(\delta)$ and $\delta$ have disjoint representatives that bound a subsurface of genus one.  Applying  $\left(T_{\alpha_2} \cdots T_{\alpha_m}\right)$ to the pair $(T_{\alpha_1}T_{\beta_1}^{-1}(\delta),\delta)$ and using the fact that each $\alpha_i$ with $i > 1$ is disjoint from $\delta$, we conclude that there are representatives of $T_AT_B^{-1}(\delta)$ and $\delta$  that are disjoint and bound a subsurface of genus one, as desired.
\end{proof}

\begin{lemma}
\label{lemma:boundary}
Fix some $g \geq 3$.  Let $M$, $\Sigma$, and $\C$ be the mapping torus, fiber, and fibered cone corresponding to $f_g$.  There are surfaces $\Sigma_0$ and $\Sigma_1$ of genus two in $M$ with $[\Sigma_0]$ and $[\Sigma_1]$ in $\bar \C$ and so that  $[\Sigma_0]$, $[\Sigma_1]$, and $[\Sigma]$ are linearly independent.
\end{lemma}

\begin{proof}

For each $g \geq 3$ we can find a curve $\delta$ in $S_g$ that---together with $A_g$ and $B_g$---satisfies the hypotheses of Lemma~\ref{lemma:boundary prelim}.  In fact, we can find two such curves, $\delta_0$ and $\delta_1$, and we can choose them so that their homology classes are linearly independent:

\bigskip

\labellist
\small\hair 2pt
\pinlabel {$\delta_1$} [ ] at 150 45
\pinlabel {$\delta_0$} [ ] at 80 45
\endlabellist
\centerline{\includegraphics[scale=1]{gammas4}}

\vspace*{2ex}

\noindent(there are many others!).  By Lemma~\ref{lemma:boundary prelim}, $f_g(\delta_i)$ and $\delta_i$ have disjoint representatives that bound a subsurface $\Xi_i$ of genus one for each $i\in\{0,1\}$.  As described in a previous paper by the second and third authors \cite[Proof of Lemma 5.1]{LM}, we can use $\Xi_0$ and $\Xi_1$ to construct surfaces $\Sigma_0$ and $\Sigma_1$ in $M$ that have genus two and are transverse to the suspension flow on $M$ associated to $f_g$; the idea is to drag $\delta_i \subset \Xi_i$ in the direction of the suspension flow of $M$ until it meets the fiber again for the first time, at which point it coincides with $f_g(\delta_i)$ and hence gives the desired closed surface.  Because the $\Sigma_i$ can be made transverse to the suspension flow, the homology classes $[\Sigma_0]$ and $[\Sigma_1]$ both lie in the closure $\bar {\C}$ (see Theorem~\ref{T:thurston}).

Finally, we prove that $[\Sigma_0]$, $[\Sigma_1]$, and $[\Sigma]$ are linearly independent.  Since $\kappa(f_g) = 2g$, the inclusion of $\Sigma \to M$ induces an injection $H_1(\Sigma) \to H_1(M)$.   As the cap product with $[\Sigma]$ is a linear function $H_2(M) \to H_1(M)$ and is given by oriented intersection with $\Sigma$, we have $[\Sigma]  \smallfrown [\Sigma] = 0$, $[\Sigma_0] \smallfrown [\Sigma] = [\delta_0]$, and $[\Sigma_1] \smallfrown [\Sigma] = [\delta_1]$.  It then follows from the fact that $[\delta_0]$ and $[\delta_1]$ are linearly independent that $[\Sigma]$, $[\Sigma_0]$, and $[\Sigma_1]$ are linearly independent.
\end{proof}

Our next goal is to give an analogue of Lemma~\ref{lemma:boundary} for $f_2$ and $f_2'$.  We again require a preliminary lemma.

\begin{lemma}
\label{lemma:prelim}
Let $\delta$ be a nonseparating simple closed curve in $S_2$, and let $\alpha$ be an essential separating simple closed curve in $S_2$ that intersects $\delta$ essentially in two points.  Let $\tilde S_2$ be the infinite cyclic cover of $S_2$ corresponding to $\delta$.  There is a subsurface $\tilde \Sigma_0 \subseteq \tilde S_2$ with genus one and with two boundary components that are connected components of the preimages of $\delta$ and (a representative of) $T_\alpha(\delta)$, respectively.
\end{lemma}

\begin{proof}

Up to homeomorphism, the curves $\delta$ and $\alpha$ are arranged in $S_2$ as in the left-hand side of the following figure:

\vspace*{1ex}

\labellist
\small\hair 2pt
\pinlabel {$\alpha$} [ ] at 37 44
\pinlabel {$\delta$} [ ] at 105 28
\pinlabel {$\tilde \delta_0$} [ ] at 176 60
\pinlabel {$\tilde \delta_1$} [ ] at 213 60
\pinlabel {$\tilde \delta_2$} [ ] at 251 60
\pinlabel {$\tilde \alpha_0$} [ ] at 168 37
\pinlabel {$\tilde \alpha_1$} [ ] at 204 37
\pinlabel {$\tilde \alpha_2$} [ ] at 240 37
\endlabellist
\begin{center}
\noindent \includegraphics[scale=1]{immersed}
\end{center}
and their preimages $\tilde \delta$ and $\tilde \alpha$ in $\tilde S_2$ are arranged as in the right-hand side.  Each component of $\tilde \delta$ is separating in $\tilde S_2$ and there is an induced ordering of the components; as such, we label the components as $\{\tilde \delta_i \mid i \in \Z\}$.  Each component of $\tilde \alpha$ intersects exactly one $\tilde \delta_i$ and so the components of $\tilde \alpha$ inherit labels $\tilde \alpha_i$ as in the figure.

Since the covering map $\tilde S_2 \to S_2$ restricts to a homeomorphism on each component of $\tilde \alpha$, the Dehn twist  $T_{\alpha}$ lifts to a multitwist $T_{\tilde \alpha}$ of $\tilde S_2$.  The preimage of $T_{\alpha}(\delta)$ in $\tilde S_2$ is $T_{\tilde \alpha}(\tilde \delta)$.  We claim that $T_{\tilde \alpha}(\tilde \delta_1)$ bounds a subsurface $\tilde \Sigma_0$ of genus one with $\tilde \delta_0$ (assuming $T_\alpha$ was taken to have support on a small regular neighborhood of $\alpha$).  Indeed, we can write the pair $(\tilde \delta_0, T_{\tilde \alpha}(\tilde \delta_1))$ as $(\tilde \delta_0, T_{\tilde \alpha_1}(\tilde \delta_1))$ since $\tilde \alpha_1$ is the only component of $\tilde \alpha$ that intersects $\tilde \delta_1$ and if we apply $T_{\tilde \alpha_1}^{-1}$ to this pair, we obtain the pair $(\tilde \delta_0, \tilde \delta_1)$, which obviously bounds a surface $\tilde \Sigma_0'$ of genus one.  Since $T_{\tilde \alpha_1}$ is a homeomorphism the surface $\tilde \Sigma_0 = T_{\tilde \alpha_1}(\tilde \Sigma_0')$ has genus one.  By construction, the two boundary components of $\tilde \Sigma_0$ are connected components of the preimages of $\delta$ and $T_{\alpha}(\delta)$.
\end{proof}

In Lemma~\ref{lemma:prelim}, it is possible to replace $S_2$ with $S_g$, in which case $\tilde \Sigma_0$ becomes a surface of genus $g-1$ with two boundary components.

\begin{lemma}
\label{lemma:boundary2}
Let $f$ be either $f_2$ or $f_2'$.  Let $M$, $\Sigma$, and $\C$ be the associated mapping torus, fiber, and fibered cone.
There is a surface $\Sigma_0$ of genus two in $M$ so $[\Sigma_0]$ is linearly independent from $[\Sigma]$.
\end{lemma}

\begin{proof}

For the case of $f_2$ let $\delta$ be the nonseparating simple closed curve in $S_2$ shown in the following figure:

\vspace*{2ex}

\labellist
\small\hair 2pt
\pinlabel {$\delta$} [ ] at 80 42
\endlabellist
\centerline{\includegraphics[scale=1]{gammas3}}

\vspace*{2ex}

\noindent and for the case of $f_2'$ let $\delta$ be either of the two nonseparating curves in $B_2'$.  Let $\alpha$ denote either the curve in $A_2$ or $A_2'$, according to our choice of $f$.  Since $\delta$ is disjoint from $B_2$ and $B_2'$ we have $f(\delta) = T_\alpha(\delta)$.  Let $\tilde S_{2}$ be the infinite cyclic cover of $S_{2}$ corresponding to $\delta$ and let $p : \tilde S_{2} \to S_{2}$ be the covering map.   Since $\alpha$ is a separating simple closed curve that intersects $\delta$ essentially in two points, Lemma~\ref{lemma:prelim} provides a subsurface $\tilde \Sigma_0$ of genus one and with two boundary components that are connected components of the preimages of $\delta$ and a representative of $T_\alpha(\delta)$.   We label the preimages of $\alpha$ and $\delta$ in $\tilde S_{2}$ as in Lemma~\ref{lemma:prelim}, and as in the proof of that lemma, we can take $\tilde \Sigma_0$ to lie between $\tilde \delta_0$ and $T_{\tilde \alpha}(\tilde \delta_1)$.  Our goal is to build a map $\psi : \tilde \Sigma_0 \to M$ with image an embedded, smooth closed surface of genus two transverse to the suspension flow.

We claim there is a smooth function $\psi_1 : \tilde \Sigma_0 \to \R$ with the following properties:
\begin{enumerate}
 \item $\psi_1|\tilde \delta_0=0$ and $\psi_1|T_{\tilde \alpha}(\tilde \delta_1) = 1$,
 \item $\psi_1$ is constant on a neighborhood of $\partial \tilde \Sigma_0$,
 \item if $x,y \in \tilde \Sigma_0$ differ by a nontrivial covering transformation of $\tilde S_{2}$ then $\psi_1(x) \neq \psi_1(y)$.
\end{enumerate}
To prove the claim, we first note that there is a function $\psi_2 : \tilde S_{2} \to \R$ with the following properties:
\begin{enumerate}
 \item $\psi_2|\tilde \delta_i=i$,
 \item $\psi_2$ is constant on a neighborhood of $\tilde \delta$,
 \item if $x$ and $y$ in $\tilde S_{2}$ differ by a nontrivial covering transformation, then $|\psi_2(x) - \psi_2(y)| \geq 1$, and
 \item $\psi_2(\tilde \alpha_1) \subseteq  (1/2,3/2)$.
\end{enumerate}
Such a function can be obtained by equivariantly perturbing the obvious (horizontal) height function on $\tilde S_{2}$, using the picture in the proof of Lemma~\ref{lemma:prelim}.

The function $\psi_1$ is then the restriction to $\tilde \Sigma_0$ of $\psi_2 \circ T_{\tilde \alpha_1}^{-1}$.  The only nontrivial property of $\psi_1$ to check is the third one.  By the fourth property of $\psi_2$ we can choose a closed neighborhood $A$ of $\tilde \alpha_1$ with $\psi_2(A) \subseteq  (1/2,3/2)$, and we can choose $T_{\tilde \alpha_1}^{-1}$ to be supported in this neighborhood.  It follows that $|\psi_1(x) - \psi_2(x)| < 1$ for all $x \in \tilde \Sigma$.  Combining this with the third property of $\psi_2$, we obtain the third desired property of $\psi_1$, finishing the proof of the claim.

Write $M$ as $S_2 \times [0,1] /\!\!\sim$ where $(x,0) \sim (f(x),1)$ and let $\pi : S_{2} \times [0,1] \to M$ denote the projection.  By the claim, we can define $\psi : \tilde \Sigma_0 \to M$ by
\[ \psi(x) = \pi(p(x),\psi_1(x)).\]
The image is a smooth, embedded surface $\Sigma_0$ of genus two since the two boundary components of $\tilde \Sigma_0$ are identified in the image.  We claim that $\Sigma_0$ is transverse to the suspension flow.  Indeed, the only way this could fail would be if $p$ were not a submersion; but $p$ is a covering map and hence is a submersion.  It follows that $[\Sigma_0]$ lies in $\bar \C$ (Theorem~\ref{T:thurston}(4)).  The linear independence is proven in the same way as in Lemma~\ref{lemma:boundary}.
\end{proof}

\p{Bounds on entropy in a fibered cone} The next lemma will make use of the following theorem of Fried \cite{Fr0,Fr}.

\begin{theorem}[Fried]
\label{theorem:fried}
Let $\C$ be a fibered cone for a fibered 3-manifold.  There is a strictly convex continuous function $\h \colon \C \to \R$ with the following properties:
\begin{enumerate}
\item for all $t > 0$ and $u \in \C$, we have $\h(tu) = \frac 1t \h(u)$;
\item for every primitive integral $u \in \C \cap H_2(M)$, the entropy of the associated monodromy is $\h(u)$; and
\item as $u \to \partial \C$ within $\C$, we have $\h(u) \to \infty$.
\end{enumerate}
\end{theorem}
See also \cite{Mc} for another proof and for finer properties of the function $\h$.

We will also make use of the normalized entropy function
\[ \bar \h (x) = \|x\| \h(x). \]
By the first property of $\h$ in Theorem~\ref{theorem:fried}, the function $\bar \h$ is constant on rays through 0.  Therefore, as $\h$ is strictly convex, the function $\bar \h$ is convex.

\begin{lemma}
\label{lemma:convex}
Let $\C$ be a fibered cone for a mapping torus $M$.  If $u \in \C$ and $v \in \bar \C$, then $\h(u+v) < \h(u)$.
\end{lemma}

\begin{proof}

Set $h_0 = \h(u)$.  By properties (1) and (2) of $\h$ in Theorem~\ref{theorem:fried}, the region $\h^{-1}((0,h_0])$ is convex.  The point $u$ is contained in the boundary, the level hypersurface $\h^{-1}(h_0)$.  By properties (1) and (3) of $\h$ in Theorem~\ref{theorem:fried}, $\h^{-1}(h_0)$ is properly embedded and asymptotic to $\partial \C$; see also \cite{Mc}.

The intersection of $\h^{-1}((0,h_0])$ with the plane $P \subset H^1(M)$ spanned by $u$ and $v$ is a convex region in $P$ bounded by $\h^{-1}(h_0) \cap P$, which we think of as a parameterized curve.  By the previous paragraph, as the parameter tends to $\pm \infty$ the slopes limit to the slopes of the two rays of $\partial \bar {\C} \cap P$:

\bigskip

\centerline{\includegraphics[scale=.5]{convex}}

\noindent It follows that $u + \bar {\C} \cap P$ (and hence $u+v$) is contained in $\h^{-1}((0,h_0])$.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{P:homology_bound_gen 2}]

For each choice of field $\F$ and each choice of $g$ and $k$ with $g \geq k/2$, we need to construct $f_{g,k} \in \Mod(S_g)$ with $\kappa(f_{g,k};\F) \geq k$ and
\[h(f_{g,k})  \leq \twoentropy \left(\frac{k + 1}{2g-2}\right ) .\]
Since $\kappa(f) \leq \kappa_\F(f)$ for any field $\F$, it suffices to consider the case $\F = \R$.

We can dispense with the case $k=0$:  we already stated in the introduction that $L(g,0) < \log(\varphi^4)/(2g-2)$ which is to say that for $g \geq 2$, there is an $f_{g,0} \in \Mod(S_g)$ with $h(f_{g,0}) < \log(\varphi^4)/(2g-2) < \twoentropy/(2g-2)$.

For $k \geq 2$ even, we will prove something stronger than what is required: we will show that there is an $f_{g,k} \in \Mod(S_g)$ with $\kappa(f_{g,k}) = k$ and
\[h(f_{g,k})  \leq \twoentropy \left(\frac{k}{2g-2}\right ) .\]
The case of $k$ odd follows immediately from this.  Indeed, for $k$ odd, we can then set $f_{g,k} = f_{g,k+1}$.  For then $\kappa(f_{g,k}) = k+1 \geq k$ and $h(f_{g,k}) < \twoentropy\,(k+1)/(2g-2)$. Thus, for the remainder of the proof, we take $k$ to be even.

Let $g_0=k/2$ if $k \geq 4$ and let $g_0=2$ if $k=2$.  Also let $f = f_{g_0}$ if $k \geq 4$ and let $f=f_2'$ if $k=2$.  Let $M$, $\Sigma$, and $\C$ be the associated mapping torus, fiber, and fibered cone.

We can firstly set $f_{g_0,k} = f$, as by Lemma~\ref{lemma:dil} we have
\[ (2g_0-2) h(f) < 2g_0 \cdot \entropy \leq  \twoentropy \cdot k.\]
Assume now that $g > g_0$.  If $g_0 \geq 3$, let $\Sigma_0$ and $\Sigma_1$ be the surfaces in $M$ promised by Lemma~\ref{lemma:boundary}, and if $g_0=2$, let $\Sigma_0$ be the surface promised by Lemma~\ref{lemma:boundary2}; for convenience we set $[\Sigma_1]=0$ in the latter case.  It follows from Lemmas~\ref{lemma:dil} and~\ref{lemma:convex} that $\h([\Sigma])$, $\h([\Sigma] + g_0 \cdot [\Sigma_0])$, and $\h([\Sigma] + [\Sigma_1])$ are all bounded above by $6 \log(2)$.

We would like to find a primitive integral class $x \in \C$ that has $\|x\| = 2g-2$ and lies in the cone on the convex hull of $[\Sigma]$, $[\Sigma] + g_0 \cdot [\Sigma_0]$, and $[\Sigma] + [\Sigma_1]$.  To do this, we write
\[ 2g-2 = \ell(2g_0-2) + 2r \]
where $\ell > 0$ and $0 < r < g_0$.  If $r$ and $\ell$ are relatively prime we take
\begin{align*}
x &= \ell [\Sigma]  + r [\Sigma_0] \\
&= \left(\ell - \frac{r}{g_0}\right)[\Sigma] + \left(\frac{r}{g_0}\right)\left( [\Sigma] + g_0 \cdot [\Sigma_0]   \right)
\end{align*}
and if $r$ and $\ell$ are not relatively prime then $r,\ell \geq 2$ and we take
\begin{align*}
x &= \ell [\Sigma]  + (r-1) [\Sigma_0] + [\Sigma_1] \\
& = \left(\ell - 1 - \frac{r-1}{g_0}\right)[\Sigma] + \left(\frac{r-1}{g_0}\right)(    [\Sigma]  + g_0 \cdot [\Sigma_0]) + ([\Sigma] + [\Sigma_1]).
\end{align*}
Note that in the latter case $r \geq 2$ implies $g_0 \geq 3$, and so $[\Sigma_1]\neq 0$.
By Theorem~\ref{T:thurston}(2) we have $\|x\| = 2g-2$ in either case.  The class $x$ is primitive because $[\Sigma]$ and $[\Sigma_0]$, and $[\Sigma_1]$ are primitive and linearly independent when $g_0 \geq 3$ (Lemma~\ref{lemma:boundary}) and  $[\Sigma]$ and $[\Sigma_0]$ are primitive and linearly independent for $g_0=2$ (Lemma~\ref{lemma:boundary2}).

By Theorem~\ref{T:thurston} the class $x$ is represented by a connected fiber for some fibration of $M$; denote the monodromy by $f_{g,k}$.  Since $\|x\|=2g-2$ the genus of this fiber is $g$, so $f_{g,k} \in \Mod(S_g)$.  Since $b_1(M)=k+1$ (Lemma~\ref{lemma:kappa}) we have $\kappa(f_{g,k}) = k$.

It remains to bound the entropy $h(f_{g,k})$.  Since $\bar \h$ is convex and constant on rays, $\bar\h(x)$ is bounded above by the values of $\bar\h$ on $[\Sigma]$,  $[\Sigma] + g_0\cdot[\Sigma_0]$, and $[\Sigma] + [\Sigma_1]$.  Applying this fact,  Lemma~\ref{lemma:convex}, and Theorem~\ref{T:thurston}(2), we have:
\begin{align*}
(2g-2) h(f_{g,k}) = \bar \h(x)  & \leq \max \{ \bar\h([\Sigma]) , \bar\h([\Sigma] + g_0 \cdot [\Sigma_0]),\bar\h([\Sigma] + [\Sigma_1]) \} \\
& \leq \max\{ \| [\Sigma] \|,\| [\Sigma] + g_0 \cdot [\Sigma_0] \|,\| [\Sigma] + [\Sigma_1] \| \} \\&   \qquad \cdot  \max\{ \h([\Sigma]), \h([\Sigma] + g_0 \cdot [\Sigma_0]),\h([\Sigma] + [\Sigma_1]) \} \\
& \leq ((2g_0-2) +2g_0) \cdot h(f) \\
& < 4g_0 \cdot h(f).
\end{align*}
When $k \geq 4$, we have $g_0=k/2$ and $h(f) < \entropy$ (Lemma~\ref{lemma:dil}) and so $h(f_{g,k}) \leq \twoentropy\cdot k/(2g-2)$, as required.  And when $k=2$ we have $g_0=k$ and $h(f) < 2$ (Lemma~\ref{lemma:dil}) and so $h(f_{g,k}) < (4k \cdot 2)/(2g-2) < 12 \log(2) \cdot (3/2g-2)$, as required.
\end{proof}

\section{Counting conjugacy classes}
\label{sec:count}

In this section we prove Theorem~\ref{T:homology_count}, which states that for all $k \geq 0$ there exists constants $c_1,c_2 >0$ so that the number of conjugacy classes of pseudo-Anosov $f \in \Mod(S_g)$ with $\kappa(f) = k$ and $h(f) < \entropy(k+1)/(2g-2)$ is at least $c_1 g^k - c_2$.
The proof is almost identical to the proof of the analogous theorem of the last two authors in the case where there is no restriction on $\kappa(f)$ \cite[Theorem 1.3]{LM}, and so we will refer to the proof in that paper for some of the details.

\begin{proof}[Proof of Theorem \ref{T:homology_count}]

The theorem is vacuously true for $k =0$, so we assume $k  > 0$.  First suppose $k \geq 2$ is even.  If $k \geq 4$, then set $g_0 = k/2$ and let $f = f_{g_0} \colon S_{g_0} \to S_{g_0}$ be the pseudo-Anosov mapping class constructed in Section~\ref{S:examples}.  If $k = 2$, then set $g_0=2$ and let $f = f_2' \colon S_{g_0} \to S_{g_0}$.  According to Lemmas~\ref{lemma:dil} and \ref{lemma:kappa}, $h(f) < \entropy$ and $\kappa(f) = k$.

Let $M$, $\Sigma$, and $\C$ be the mapping torus, fiber, and fibered cone corresponding to $f$, and let $\h \colon \C \to \R$ be the function from Theorem~\ref{theorem:fried}.  By Theorem~\ref{T:thurston}, the restriction of the Thurston norm to $\C$ is the restriction of an integral linear functional $L$ on $H_2(M)$.

Choose any compact neighborhood $K \subset L^{-1}(1) \cap \C$ of $[\Sigma]/\|[\Sigma]\|$ on which the function $\bar\h(x) = \|x\| \h(x)$ is bounded by $6 \log(2) \cdot (2g_0-2) < 6\log(2) \cdot (k+1)$.  Since $\bar\h$ is invariant under scale such a $K$ exists and for the same reason  $\bar\h$ is bounded by $\entropy(k+1)$ on $\R_+ \cdot K$, the cone over $K$.

For every $g \geq g_0$ we set
\[
\Omega_{g,k} = \{ [\Sigma] \in (2g-2) \cdot K \mid [\Sigma] \mbox{ is primitive integral}  \}.
\]
Each element $[\Sigma]$ of $\Omega_{g,k}$ is represented by a fiber of $M$ of genus $g$ for which the monodromy $f_\Sigma \colon \Sigma \to \Sigma$ satisfies $h(f_\Sigma) \leq \entropy \cdot (k+1)/(2g-2)$, as in the statement of the theorem.  We would like to estimate from below the size of $\Omega_{g,k}$.  Since $L$ is integral, we can write $H_2(M;\Z) \cong \Z^k \oplus \Z$, where the first summand is the kernel of the restriction of $L$.  This decomposition extends to a decomposition of $H_2(M)$ as $\R^k \oplus \R$.  By Lemmas~\ref{lemma:boundary} and~\ref{lemma:boundary2} we have $L(H_2(M;\Z)) = 2\Z$.

Let $K_0$ denote the image in $\R^k$ of $K$ under orthogonal projection.  By the previous paragraph, we can identify the integral points of $(2g-2) \cdot K$ with $\Z^k \cap (2g-2) \cdot K_0$ (where the scaling is done in $\R^k$).  As the volume of $(2g-2) \cdot K_0$ grows like $g^k$, it follows that the number of integral points in $(2g-2) \cdot K$ is at least $c_1''g^k - c_2''$ for some $c_1'',c_2'' > 0$ (we must subtract $c_2''$ because, for instance, the set is empty for small values of $g$).   Then by noting that the number of primitive elements make up a definite fraction of the integral elements, we can deduce that for all $g \geq 2$, $|\Omega_{g,k}|$ is bounded below by $c_1'g^k - c_2'$ for some $c_1',c_2' > 0$; see \cite{LM} for the details.

If two points of $\Omega_{g,k}$ have monodromies that are conjugate in $\Mod(S_g)$, then there is a homeomorphism of $M$ taking one fiber to the other.  By Mostow rigidity, such a homeomorphism is homotopic to an isometry of $M$ with respect to its hyperbolic metric.  It follows that the number of conjugacy classes in $\Mod(S_g)$ represented by elements of $\Omega_{g,k}$ is at least $(c_1'g^k - c_2')/N$, where $N$ is the order of the isometry group of $M$.  Setting $c_1 = c_1'/N$ and $c_2 = c_2'/N$ completes the proof in the case of $k$ even.

Now suppose that $k \geq 5$ is odd.  The argument is almost the same as the case when $k$ is even.  Let $g_0 = (k+1)/2$.  Let $f'$ be the mapping class defined in the same way as $f_{g_0}$ except that we leave out one nonseparating curve from the construction, namely, any nonseparating curve disjoint from $\delta_0$.

By calculations similar to those made in Lemmas~\ref{lemma:dil} and~\ref{lemma:kappa} the mapping class $f'$ is pseudo-Anosov with $\kappa(f') = k$ and with
\[
h(f') \leq h(f) < \entropy (2g_0-2) < \entropy (k+1).
\]
The proof of Lemma~\ref{lemma:boundary} applies and so there is a surface of genus two in the boundary of the fibered cone corresponding to $f'$.  We may apply the argument given above for the case of $k$ even, with $f$ replaced by $f'$.

\medskip

For the cases of $k=3$ and $k=1$, we need slightly different examples from the ones previously given in $\Mod(S_2)$, namely, the ones given by the product of two positive multitwists about the following pairs of multicurves:

\vspace*{2ex}

\labellist
\small\hair 2pt
\endlabellist
\centerline{\includegraphics[scale=1]{breadstick4prime}}

\vspace*{2ex}

For the pair of multicurves on the left-hand side we can compute $h(T_AT_B) < \log(34)$ and $\kappa(T_AT_B) = 3$, and for the pair on the right-hand side we can compute $h(T_AT_B) < \log(9)$ and $\kappa(T_AT_B) = 1$.   As the proof of Lemma~\ref{lemma:boundary2} applies in both cases, we can repeat the argument from above to complete the proof.
\end{proof}

As promised in the introduction, we now explain how to promote our Main Theorem to show that the minimal entropy of a pseudo-Anosov element of $\Mod(S_g)$ fixing a subspace of $H_1(S_g;\F)$ of dimension exactly equal to $k$ is comparable to $(k+1)/g$.  For the lower bound there is nothing to do.  For the upper bound and $k>0$ even the examples given in the proof of Proposition~\ref{P:homology_bound_gen 2} have $\kappa_\F = k$, and so there is again nothing to do.   For every odd $k$, we must construct a mapping class as in that proof, except with $\kappa_\F$ equal to $k$ instead of $k+1$.  To do this we simply replace the $f_{g_0}$ in the proof of Proposition~\ref{P:homology_bound_gen 2} with their counterparts described in the proof of Theorem~\ref{T:homology_count}.  The entropies of the latter are bounded above by $\entropy$ since they are obtained by forgetting Dehn twists from the $f_{g_0}$, and they have $\kappa_\F = k$.

For $k=0$, the techniques of this paper do not apply, since the mapping torus associated to $f$ with $\kappa(f) = 0$ has first betti number equal to 1, and so there are no fibered cones.  However, one can show Penner's examples \cite[p. 448]{Pe} giving the required upper bound have $\kappa_\F=0$ for all $g$ and $\F$.

redpajama_set_name: RedPajamaArXiv