text:
\section{Introduction}

\noindent
Our aim is to suggest a shorter and we hope considerably simpler version of the proof for Higman's remarkable embedding theorem:

\begin{Theorem}[Theorem 1 in~\cite{Higman Subgroups of fP groups}]
\label{TH Higman 1}
A finitely generated group can be embedded in a finitely presented group if and only if it is recursively presented.
\end{Theorem}

Recursive presentation for a group $G$ presumes that there is a presentation $G=\langle x_1, \ldots, x_n \mathrel{|} r_1, r_2, \ldots\, \rangle$ such that the set of defining relations $\{r_1, r_2, \ldots\}$ is recursive enumerable, i.e., it is the range of some recursive function.
This result establishes deep connections between the logical notion of recursion and group theory.
For many of its applications we refer to
Ch.~12 of \cite{Rotman},
Ch.~IV of \cite{Lyndon Schupp},
Ch.~I, IV, VI in \cite{Baumslag Topics in Combinatorial}, and to newer work of Ol'shanskii and Sapir \cite{Ol'shanski Sapir The conjugacy problem}.

\begin{comment}
\subsection{\color{DarkBlue}The main steps of the proof in ~\cite{Higman Subgroups of fP groups}}
\label{SU The main steps of the proof of Higman}
\end{comment}

\subsection{The main steps of Higman's proof}
\label{SU The main steps of Higman's proof}
\cite{Higman Subgroups of fP groups} starts by Kleene's formal characterization of partial recursive functions on the set of non-negative integers
as the class of
functions that can be obtained from the {zero},
 {successor}
and
{identity} functions
using the operations of  composition, primitive recursion, and minimization (see \cite{Davis, Rogers} or the newer text \cite{Boolos Burgess Jeffrey}).
Let $\Z$ be the set of all integers, and $\mathcal E$ be the set of all functions $f : \Z \to \Z$ with finite supports in the sense that $f(i)=0$ for all but finitely many integers $i\in \Z$. To each such function $f\in \E$ one can constructively assign a unique non-negative  G\"odel number.
This makes $\mathcal E$ an effectively enumerable set, and one may put into correspondence to each subset $\B$ of $\E$ the respective set of non-negative integers corresponding to the functions $f\in \B$.
Then $\B$ is \textit{recursively enumerable}, if that set is the range of a recursive function.
Next, Section 2 in \cite{Higman Subgroups of fP groups} gives a different characterization  for recursively enumerable subsets of $\E$: two initial subsets $\Zz$ and $\S$ of $\E$, and a series of operations on subsets of $\E$ are introduced (see \eqref{EQ Higman operations} in \ref{SU The Higman operations on subsets} below), and  Theorem 3 states that $\B$ is recursively enumerable if and only if it can be obtained from $\Zz$ and $\S$ by operations \eqref{EQ Higman operations}.
Only after these preparations the  group-theoretical argument starts in~\cite{Higman Subgroups of fP groups}. The key concept of \textit{benign subgroup} is introduced (see \ref{SU Definition and main properties of benign subgroups}), and to each $\B$ a specific subgroup $A_\B$ of the free group $G=\langle a,b,c \rangle$ of rank $3$ is put into correspondence. The main result of Section 3 and of Section 4 in \cite{Higman Subgroups of fP groups} is:

\begin{Theorem}[Theorem 4 in~\cite{Higman Subgroups of fP groups}]
\label{TH Higman 4}
The subset $\mathcal B$ of $\mathcal E$ is recursively enumerable if and only if $A_{\mathcal B}$ is a benign subgroup in $G=\langle a,b,c \rangle$.
\end{Theorem}

Then in final brief Section 5 the above theorem is generalized:
a subgroup of \textit{any} finitely generated free group is recursively enumerable if and only if it is benign. The ``Higman Rope Trick'', as it is often
called~\cite{Higman rope trick}, closes the proof by showing how a recursively presented group can be embedded in a finitely presented group.
Here we suggest a shorter proof for Theorem~\ref{TH Higman 4}, and our sections \ref{SE The main properties and basic examples of benign subgorups}, \ref{SE The proof of Theorem} roughly correspond to sections 3, 4 in \cite{Higman Subgroups of fP groups}.

\subsection{Comparison of the current modification with~\cite{Higman Subgroups of fP groups}}
Leaving aside minor changes, here are the main modifications done to Higman's original construction.

We construct a very different group to show that
the set of all subsets $\mathcal B \subseteq \E$, for which the subgroup $A_{\mathcal B}= \langle a_f \;|\; f \in \mathcal B\, \rangle$ is benign, is closed under Higman operation $\omega_m$. In \cite{Higman Subgroups of fP groups} this is done by the group $M *_{\langle a,b,c \rangle} K$ built using Lemma 4.10, Lemma 3.9 and Lemma 3.10, which perhaps form the most complicated part in \cite{Higman Subgroups of fP groups}. Compare them with the part ``\textit{$\BB$ is closed under  $\omega_m$}\!''  in Section~\ref{SE The proof of Theorem} below.

Three lemmas 3.2--3.4 on homomorphisms are extensively used in~\cite{Higman Subgroups of fP groups} to study subgroups in free constructions. We obtain similar results using simple combinatorics on words, and  trivial observations made in \ref{SU The conjugates collecting process}. Compare, for instance,
Lemma 3.8 and Lemma 3.9 of ~\cite{Higman Subgroups of fP groups} with direct proofs in examples listed in \ref{SU Basic examples of benign subgroups} below.

We accompany all concepts by examples, and give much more detailed proofs. In particular,
we do prove the part
\textit{``These two conditions are easy, but a little tedious, to check, and this is left to the reader''} on p.\,472 in  \cite{Higman Subgroups of fP groups}, and
we do give a proof for
Lemma~\ref{LE subgroups in amalgamated product} mentioned in ~\cite{Higman Subgroups of fP groups} as \textit{``obvious from the normal form theorem''}.
In fact, we could shorten the proofs even more by evolving some of our wreath product methods \cite{Subnormal embedding theorems}--\cite{Subvariety structures}. However, we intentionally keep  technique within free products with amalgamations and  HNN-extensions to preserve Higman's original idea of investigation of recursion via free constructions.

\subsection{Other proofs for Higman's embedding theorem}

Higman's original proof is very complicated, and this motivated others to suggest alternative proofs for Theorem~\ref{TH Higman 1}.
Lindon and Schupp present in Section IV\,\!.\,\!7 of \cite{Lyndon Schupp} a proof related to Valiev's approach \cite{Valiev}.
Solving Hilbert's Tenth Problem \cite{Matiyasevich Diophantine} established  that a subset of $\Z^n$ is recursively enumerable if and only if it is Diophantine.
The proof in \cite{Lyndon Schupp}  uses this Diophantine characterization, i.e., it relies on a ``third party'' result  (which itself has a complicated proof).

A proof reflecting  Aanderaa's  work \cite{Aanderaa} is given by Rotman in Ch.\,12 of \cite{Rotman}. It applies the auxiliary construction  interpreting Turing machines via semigroups, and also uses the Boone-Britton group  \cite{Boone Certain simple unsolvable problems, Britton The word problem}. Group diagrams allow \cite{Rotman} to shorten the proof of \cite{Aanderaa}.

Besides the mentioned two well known textbooks, other proofs can be found in the Appendix of Shoenfield's textbook on logic  \cite{Shoenfield}, in the article of Adyan and Durnev \cite{Adyan Durnev}, etc. Typically, they relay on auxiliary  constructions, such as those  built to show undecidability of the word problem in semigroups \cite{Post Recursive unsolvability,
Markov impossibility of certain algorithms} and in  groups
\cite{Novikov on algorithmic undecidability,
Boone Certain simple unsolvable problems,
Britton The word problem}.

\smallskip
This comparison stresses Higman's more straightforward approach of \textit{``mimicking recursion''} by some elegant group-theoretical constructions.
And a motivation of our modification is to emphasize that Higman's original idea still is one of the clearest approaches to handle this subject.

\subsection*{Acknowledgements}
The current work is supported by the joint grant 18RF-109 of RFBR and SCS MES RA, and by the 18T-1A306 grant of SCS MES RA.
I would like to also thank my both universities, YSU and AUA, for permanent and substantial support.
The work was announced at the International Algebraic Conference in memory of A. G. Kurosh, May 23-25, 2018, Moscow.

\section{Notations and preliminary information}

\subsection{Sets of integer-valued funations}
\label{SU Sets of integer-valued funations}

In~\ref{SU The main steps of Higman's proof} we introduced the set  $\mathcal E$ of all functions $f : \Z \to \Z$ with finite supports.
If $f(i)=0$ for any $i<0$ and $i\ge n$, we may record such a function as $f=(a_0,\ldots,a_{n-1})$ assuming $f(i)=a_i$ for $i=0,\ldots,n\!-\!1$. Say,
$f=(0,0,7,-8,5,5,5,5)$
means
$f(2)=7$, $f(3)=-8$,
$f(i)=5$ for $i=4,\ldots,7$, and
$f(i)=0$ for any $i<2$ or $i\ge 8$.
Denote by $f=(0)$ the constant zero function on $\Z$.
For a fixed integer $m$ and for an $f\in \mathcal E$  denote the functions $f_{m}^+$ and $f_{m}^-$ as follows.
$f_{m}^+(i)=f_{m}^-(i)=f(i)$ for all $i\!\neq\! m$, and
$f_{m}^+(m) = f(m) +1$,\;
$f_{m}^-(j) = f(m) -1$.
When $m$ is given by the context, we may shorten these to $f_{m}^+=f^+$ and $f_{m}^-=f^-$. Say, for the above $f$ we get $f_{5}^+\!\!=(0,0,7,-8,5,6,5,5)$, and
$f^-\!\!=(0,0,7,-8,5,5,5,4)$, provided $m=8$ is given.

\subsection{The Higman operations on subsets of $\E$}
\label{SU The Higman operations on subsets}

Define some specific subsets of $\E$:
$$
\Zz\!=\!\big\{(0)\big\},\;\;
\S\!=\!\big\{(a, a+\!1) \mathrel{|} a\!\in\! \Z\big\},\;\;
\E_m\!\!=\!\big\{
(a_0,\ldots,a_{m-1})
\mathrel{|} a_i\!\in \!\Z,\; i\!=\!0,\ldots,\!m-1\big\},
\, m\!=\!0,\!1,\ldots
$$
($\E_0$ consists of zero function only).
We will use the following  operations from \cite{Higman Subgroups of fP groups}:
\begin{equation}
\label{EQ Higman operations}
\tag{H}
\iota,\; \upsilon;\quad\quad
\rho,\; \sigma,\; \tau,\; \theta,\; \zeta,\; \pi,\; \omega_m \;\; (\text{for each $m=1,2,\ldots$})
\end{equation}
which we call \textit{Higman operations}.
For any subsets $\mathcal A, \mathcal B$ of $\E$ define
$\iota(\mathcal A,\mathcal B)=\mathcal A \cap \mathcal B$ and
$\upsilon(\mathcal A,\mathcal B)=\mathcal A \cup \mathcal B$.
The rest of Higman operations  are unary functions on the subsets of $\mathcal E$:

$f \in \rho(\mathcal A)$ iff there is a $g\in \mathcal A$ such that $f(i)=-g(i)$,

$f \!\in\! \sigma(\mathcal A)$ iff there\! is\! a\! $g\in \mathcal A$ such\! that\! $f(i)=g(i-1)$,

$f\! \in \tau(\mathcal A)$ iff there is a $g\!\in\! \mathcal A$ such that $f(0)\!=\!g(1)$, $f(1)\!=\!g(0)$ and  $f(i)\!=\!g(i)$ for $i\!\neq\! 0,1$,

$f \in \theta(\mathcal A)$ iff there is a $g\in \mathcal A$ such that $f(i)=g(2i)$,

$f \in \zeta(\mathcal A)$ iff there is a $g\in \mathcal A$ such that $f(i)=g(i)$ for  $i\neq 0$,

$f \in \pi(\mathcal A)$ iff there is a $g\in \mathcal A$ such that $f(i)=g(i)$ for  $i\le 0$,

$f \in \omega_m(\mathcal A)$ iff for every $i\in \Z$ there is a  $g = \big(f(mi),\;f(mi+1),\ldots,f(mi+m-1)\big)\in \mathcal A$. Since  the support of any $f\in \E$ is finite, either $\mathcal A$ contains the zero function, or $\omega_m(\mathcal A)=\emptyset$.

\smallskip
The more general operations $\alpha$ and $\lambda$ mentioned \cite{Higman Subgroups of fP groups} are not used here.

\begin{Example}
\label{EX simple Higman operations}
Some simple applications of Higman operations are easy to check:

$\rho(\S)
=\big\{f\in \E \mathrel{|}
f(-1)=f(0)+1,  f(i)=0 \text{ for } i\neq -1,0
\big\}
$.

Thus, $\sigma\rho(\S)=\big\{(a+\!1,a) \mathrel{|} a\!\in\! \Z\big\}=\tau(\S)$.

Since $\zeta(\Zz)=
\big\{(a) \mathrel{|} a\in \Z\big\}=
\big\{(a,0) \mathrel{|} n\in \Z\big\}$, we have $\iota\big(\!\tau(\S),\zeta(\Zz)\big)=\big\{(1,0)\big\}=\big\{(1)\big\}$.

As $\sigma \zeta (\Zz)= \big\{(0,a) \mathrel{|} a\in \Z\big\}$, we have
$\zeta (\sigma \zeta)^{2} (\Zz)=
\big\{ (a_0,a_1,a_2) \mathrel{|} a_0,a_1,a_2\in \Z\big\}
= \E_3$.

Since $\pi (\Zz)=\{f\in \E \mathrel{|}\
f(i)=0 \text{ for } i\le 0
\}$, then
$\sigma^3 \rho \pi (\Zz)=\{f\in \E \mathrel{|}\
f(i)=0 \text{ for } i\ge 3
\}$.

Clearly, $\theta (\E_4)=
\big\{ (a_0,a_2) \mathrel{|} \text{ for each }  (a_0,a_1,a_2,a_3)\in \E_4\big\}
= \E_2$.

$\omega_2\big(\!\upsilon(\S, \Zz)\big)=\{f\in \E \mathrel{|}\
f(2i+1)=f(2i)+1 \text{ \,or\,} f(2i+1)=f(2i)=0 \text{ for any } i\in \Z
\}$. Say, $(7,8,\;0,0,\; 2,3)\in \omega_2\big(\!\upsilon(\S, \Zz)\big)$ because $(7,8), (2,3)\in \S$ and $(0,0)\in \Zz$.
Also, $\omega_2(\S)=\emptyset$.

\end{Example}

\subsection{Notations for groups and homomorphisms}
\label{SU Notations for groups and homomorphisms}

Below we may use the functions  $f\in \mathcal E$ to define specific elements in free groups.
Here is a typical context: if $G=\langle a,b,c \rangle$ is a free group of rank $3$, then the elements $b_i = b^{c^i} = c_i^{-1} b\, c_i^{\vphantom8}$,\, $i\in \Z$, are generating a free subgroup  of countable rank in $G$.
For every  $f\in \E$ we can define the element
$b_f= \cdots b_{-1}^{f(-1)} b_{0}^{f(0)} b_{1}^{f(1)} \cdots$
In particular, when $f=(a_0,\ldots,a_{m-1})\in \mathcal E$, we  define $b_f= b_0^{f(0)}\cdots b_{m-1}^{f(m-1)}$\!.\,
The elements of type $a_f=a^{b_f}$ generate yet another free subgroup in $G$.

\smallskip
If $G = \langle\, X \mathrel{|} R \,\rangle$ is the presentation of the group $G$ by its generators $X$ and definning relations $R$, then for an alphabet $Y$ (disjoint from $X$) and for any set $S\subseteq F_Y$ of words on $Y$ we denote
$\langle G, Y \mathrel{|} S \,\rangle = \langle \,X \cup Y \mathrel{|}  R \cup S \,\rangle$
(the cases when $X = \emptyset$ or $S = \emptyset$ are not ruled out).

If $\varphi:G\to H$ is a homomorphism defined on any group $G=\langle a,b,\ldots \rangle$ by the images $\varphi(a)=a'$,  $\varphi(b)=b',\ldots$\; of its generators $a,b,\ldots$\,, we may for briefness refer to $\varphi$ as \textit{the homomorphism sending} $a,b,\ldots$ \;to\; $a',b',\ldots$

\subsection{Notations for free constructions}
\label{SU Notations for free constructions}

We are going to extensively use the operations of free products of groups with amalgamated subgroups and of HNN-extensions of groups by one or more stable letters.
Referring for background information to \cite{Lyndon Schupp, Bogopolski,Rotman}, here we just state the notations we use.

If the groups $G$ and $H$ have subgroups, respectively, $A$ and $B$ isomorphic under the isomorphism $\varphi : A \to B$, then we denote the (generalized) free product $\Gamma = \langle G, H \mathrel{|} a=a^{\varphi} \text{ for all $a\in A$}\, \rangle$ of $G$ and $H$ with amalgamated subgroups $A$ and $B$ by
$G*_{\varphi} \!H$. In the simplest case, when $G$ and $H$ are overgroups of the same subgroup $A$, and $\varphi$ is just the \textit{identical} isomorphism on $A$, we may prefer to write $\Gamma = G*_{A} H$.

If the group $G$ has subgroups $A$ and $B$ isomorphic under the isomorphism $\varphi : A \to B$, then we denote the HNN-extension $\Gamma = \langle G, t \mathrel{|} a^t=a^{\varphi} \text{ for all $a\in A$}\, \rangle$ of the base $G$
by the stable letter $t$
with respect to the isomorphism
$\varphi$ by
$G*_{\varphi} t$.
In case when $A=B$ and $\varphi$ is  \textit{identity} on $A$, we may write $\Gamma = G*_{A} t$.
We also use HNN-extensions with more than one stable letters. If we have the isomorphisms $\varphi_1: A_1 \to B_1,\; \varphi_2: A_2 \to B_2,\ldots$ for pairs of subgroups in $G$,  we denote the respective HNN-extension $\langle G, t_1, t_2,\ldots \mathrel{|} a_1^{t_1}=a_1^{\varphi_1}\!\!,\; a_2^{t_2}=a_2^{\varphi_2}\!\!,\,\ldots\; \text{ for all $a_1\in A_1$, $a_2\in A_2,\ldots$}\, \rangle$ by $G *_{\varphi_1, \varphi_2, \ldots} (t_1, t_2, \ldots)$.
Our usage of the \textit{normal forms} in
free constructions is closer to~\cite{Bogopolski} (see the proof for Lemma~\ref{LE subgroups in amalgamated product} and examples in \ref{SU Basic examples of benign subgroups}).

\begin{comment}
 utilizing pairs of fixed transversals $T_A$, $T_B$ for isomorphic subgroups $A,B$ in $G$ (or more than one pairs of transversals, if more then one isomorphisms are given).
Namely, we use the well known procedure of  ``scanning'' an element from the right to the left:
\;
if $g\in G$ is written as $g=uv$ with $u\in A$ and $v \in T_A$, then
$t^{-1}\!g=t^{-1}u t \cdot t^{-1} v
=\varphi(u)\, t^{-1} v$.
\;
And if $g=wh$ with $w\in B$ and $h \in T_B$, we have
$t g= t w t^{-1} \cdot t h
=\varphi(w)^{-1}\, t h$.
\end{comment}

\begin{comment}

\begin{Lemma}
In $G *_A t$ the subgroups $G$ and $G^t$ generate the amalgamated product $G *_A G^t$.
\end{Lemma}

\begin{Lemma}
Let $\alpha$ be any automorphism of the group $G$, and $\beta$ be any automorphism of the group $H$. If $\alpha$ and $\beta$ agree on a common subgroup $A$ of $G$ and $H$, then they have a common extension $\gamma$ on the free product $G*_{A} H$ with amalgamation. [Tanel verch?]
\end{Lemma}
\end{comment}

\subsection{Subgroups in free constructions}
\label{SU Subgroups in free constructions}

Lemma~\ref{LE subgroups in amalgamated product} is a slight variation of Lemma 3.1 given in \cite{Higman Subgroups of fP groups} without a proo
\footnote{\textit{A remark for the Editor and Referee}. The proof of Lemma~\ref{LE subgroups in amalgamated product} is an application  of basic facts on the normal form in free product with amalgamation. Higman omits the proof in~\cite{Higman Subgroups of fP groups}. If this proof is too basic, I can remove it here, too. On the other hand, I can add the proof to Lemma~\ref{LE subgroups in HNN-extension}, if needed.}\!.
Lemma~\ref{LE subgroups in HNN-extension} is the analog of the latter for HNN-extensions.

\begin{Lemma}
\label{LE subgroups in amalgamated product}
Let $\Gamma = G *_\varphi \!H$ be the free product of the groups $G$ and $H$ with amalgamated subgroups
$A \le G$ and $B \le H$
with respect to the isomorphism
$\varphi: A \to B$.
If $G', H'$ respectively are subgroups of $G, H$, such that
for $A'=G'\cap A$ and $B'=H'\cap B$
we have
$\varphi (A') = B'$, then for the subgroup $\Gamma'=\langle G',H'\rangle$  of $\Gamma$ and for the restriction $\varphi'$ of $\varphi$ on $A'$ we have:
\begin{enumerate}
\item $\Gamma' = G'*_{\varphi'} H'$,

\item $\Gamma' \cap A = A'$ and $\Gamma' \cap B=B'$,

\item $\Gamma' \cap G = G'$ and $\Gamma' \cap H = H'$.
\end{enumerate}
\end{Lemma}

\begin{proof}
By definition $\Gamma=(G * H)/N$ where
$N$ is the normal closure of $\{ \varphi(a)\, a^{\!-1}\! \mathrel{|} a\in A\}$.

Each element in $\Gamma'$ can be presented as
$c= c_0 \cdots c_{m-1}  c_m$
with each term
$c_i$ picked from the factors $G'$ or $H'$ (the case $m=0$ is not ruled out). By where necessary merging the factors, we may suppose any two consecutive  terms $c_i, c_{i+1}$ are picked from different factors.

\begin{comment}
Since by definition $\Gamma=(G * H)/N$ where
$N$ is the normal closure of $\{ \varphi(a)\, a^{\!-1} \mathrel{|} a\in A\}$, then $c$  clearly is in
$\big((G' * H')\,N\big)/N$.
\end{comment}
Fix a transversal $T_{A'}$ to $A'$ in $G'$, and a transversal $T_{B'}$ to $B'$ in $H'$, and apply to $c$ the ``analog'' of  procedure of bringing to a normal form. Namely,
If, say, $c_m \!\!\in G'$, write $c_m = u \, l_n$ where $u \in A'$ and  $l_n  \in T_{A'}$ (ignore the value $n$ for now).
Since $c_{m-1}\!\in H'$, then $c_{m-1}\,u=c_{m-1}  v \in H'$ for $v=\varphi(u)\in B'$, and so  $c_{m-1}  v = w \, l_{n-1}$  where $w \in B'$ and  $l_{n-1}  \in T_{B'}$.
We already have the last two terms for $c= c_0 \cdots c_{m-2}\cdot w\,  l_{n-1} l_n$.
Continuing the process we
get:
\begin{equation}
\label{EQ normal form of g}
c=l_0 \, l_1 \cdots l_{n-1} l_n
\end{equation}
where $n \le m$, \,
$l_0\in A'$ or $l_0\in B'$, and each of terms
$l_1,\ldots, l_n$ is a non-trivial element from $T_{A'}$ or  $T_{B'}$ such that no two consecutive  terms are from the same transversal.

To prove point (1) it is enough to show that \eqref{EQ normal form of g} is \textit{unique} for any $c\in \Gamma'$ (unique normal forms are one of the ways to define free products with amalgamations).
Notice that if $l_i, l_j$ from $T_{A'}$ are distinct modulo $A'$, they also are distinct modulo $A$ because $l_i,l_j$ are in $G'$, and so  $l_i^{\vphantom8}  l_j^{-1}\! \in A$ would imply
$l_i^{\vphantom8}  l_j^{-1}\!\in (G' \cap A) = A'$.
So we can choose a transversal $T_{A}$ to $A$ in $G$, containing $T_{A'}$ as its subset.
Similarly, since if $l_i, l_j$ from $T_{B'}$ are distinct modulo $B'$, they also are distinct modulo $B$, and we can choose a transversal $T_{B}$ to $B$ in $H$, containing $T_{B'}$.
Finally, we may consider $l_0$ as an element from $A$ or $B$.
Thus, \eqref{EQ normal form of g} is nothing but the normal form of $c$ in $\Gamma = G *_\varphi H$ written using $T_A$ and $T_B$.
Since it is unique,  point 1 is proved.

Points (2), (3) now follow from point (1), and from uniqueness of the normal form.
\end{proof}

\begin{Corollary}
\label{CO G*H lemma corollary}
If $\Gamma = G*_{A} H$, then in $\Gamma$:

\begin{enumerate}
\item for any  $G'\!\!\le\! G$, $H'\!\le\! H$, such that $G' \cap\, A = H' \cap\, A$\,, we have $\langle G'\!\!, H' \rangle= G' \!*_{G' \cap \;A} H'$\!,

\item for any  $G'\!\le G$, $H'\le H$, both containing $A$\,, we have $\langle G', H' \rangle= G' *_{A} H'$.
\end{enumerate}
\end{Corollary}

\begin{Lemma}
\label{LE subgroups in HNN-extension}
Let $\Gamma=G *_{\varphi} t$ be the HNN-extension of the base group $G$ by the stable letter $t$
with respect to the isomorphism
$\varphi: A \to B$
of the subgroups
$A, B \le G$.
If $G'$ is a subgroup of $G$  such that
for $A'=G'\cap\, A$ and $B'=G'\cap\, B$
we have
$\varphi (A') = B'$, then for the subgroup $\Gamma'=\langle G',t\rangle$  of $\Gamma$ and for the restriction $\varphi'$ of $\varphi$ on $A'$ we have:
\begin{enumerate}
\item $\Gamma' =G' *_{\varphi'} t$,

\item $\Gamma' \cap G = G'$,

\item $\Gamma' \cap \,A = A'$ \;and\; $\Gamma' \cap B = B'$.
\end{enumerate}
\end{Lemma}

The proof can be conducted in analogy with the proof of Lemma~\ref{LE subgroups in amalgamated product}.

\begin{Corollary}
\label{CO G*t lemma corollary}
If $\Gamma = G*_{A} t$, then in $\Gamma$:

\begin{enumerate}
\item for any subgroup $G'$ of $G$ we have $\langle G', t \rangle= G' *_{G'  \cap\; A} t$,

\item for any subgroup $G'$ of $G$ containing $A$ we have $\langle G', t \rangle= G' *_{A} t$.
\end{enumerate}
\end{Corollary}

\subsection{The ``conjugates collecting'' process}
\label{SU The conjugates collecting process}

Let $\X$ and $\Y$
be any disjoint subsets in a group $G$. It is easy to see that any element $w\in \langle \X,\Y \rangle$ can be written as:
\begin{equation}
\label{EQ elements <X,Y>}
w=u\cdot v
= x_1^{\pm v_1}
x_2^{\pm v_2}
\cdots
x_{k}^{\pm v_k}
\cdot
v
\end{equation}
with some $v_1,v_2,\ldots,v_k,\, v\in\langle \Y \rangle$, and
$x_1,x_2,\ldots,x_k \in \X$.
Indeed, since $w$ is in $\langle \X,\Y \rangle$, present $w$ as a product of elements from $\X$, from $\Y$ and of their inverses. Then by grouping where necessary some elements from $\Y$, and by adding some trivial elements we can rewrite it as:
\begin{equation}
\label{EQ preliminary step}
w=z_1^{\vphantom8} x_1^{\pm1}
z_2^{\vphantom8} \, x_2^{\pm1} z_3^{\vphantom8}
\cdots
z_k^{\vphantom8}\, x_k^{\pm1} z_{k+1}^{\vphantom8}
\end{equation}
where $x_1,\ldots,x_k \in \X$ and
$z_1,\ldots,z_{k+1} \in \langle \Y \rangle$.
For instance, if $\X =\{x_1, x_2, x_3\}$ and
$\Y =\{y_1, y_2\}$, then
$w=x_2^{-1} y_1^3  y_2^{\vphantom8} x_1^2 x_3^{\vphantom8}$ can be rewritten as
$w=z_1\, x_2^{-1} z_2^{\vphantom8} x_1 z_3^{\vphantom8} x_1
z_4^{\vphantom8}
x_3^{\vphantom8}
z_5^{\vphantom8}
$,
where
$z_1=1$,\,
$z_2=y_1^3  y_2^{\vphantom8}$,\,
$z_3=z_4=z_5=1$ are in $\langle \Y \rangle$. Next, \eqref{EQ preliminary step} can be transformed to:
$$w=z_1^{\vphantom8} x_1^{\pm1}
z_1^{-1} \!\cdot\,
z_1^{\vphantom8}
z_2^{\vphantom8} \, x_2^{\pm1}
(z_1^{\vphantom8}
z_2^{\vphantom8})^{-1}
\!\cdot\,
z_1^{\vphantom8}
z_2^{\vphantom8}
z_3^{\vphantom8}
\cdots
(z_1^{\vphantom8}\!\cdots z_k^{\vphantom8})\, x_k^{\pm1}
(z_1^{\vphantom8}\!\cdots z_k^{\vphantom8})^{-1}
z_1^{\vphantom8}\!\cdots z_k^{\vphantom8} z_{k+1}^{\vphantom8},
$$
which is \eqref{EQ elements <X,Y>} for
$v_1^{\vphantom8}\!=z_1^{-1}$\!\!\!, \;
$v_2^{\vphantom8}\!=(z_1z_2)^{-1}$\!\!\!\!, \;
$v_3^{\vphantom8}\!=(z_1 z_2 z_3)^{-1}$\!\!\!\!, \, $\ldots$
$v_k^{\vphantom8}\!=(z_1 \cdots z_k)^{-1}$\!\!\!\!, \;
$v\!=z_1^{\vphantom8}\!\cdots\; z_k^{\vphantom8} z_{k+1}^{\vphantom8}$.

In particular, setting $\X=\{x\}$ and $\Y=\{y\}$ in a 2-generator group $G=\langle x, y\rangle$ we can present any element $w \in G$ as a product
of some conjugates of $x$ and of some power of $y$:
\begin{equation}
\label{EQ elements from <x,y>}
w=x^{\pm y^{n_1}}\!x^{\pm y^{n_2}}\!\!\cdots\, x^{y^{\pm n_s}}\!\!\cdot y^k=\!u\cdot v\,.
\end{equation}

\section{The main properties and basic examples of benign subgroups}
\label{SE The main properties and basic examples of benign subgorups}

\subsection{Definition and main properties of benign subgroups}
\label{SU Definition and main properties of benign subgroups}
The central group-theoretical notion used in \cite{Higman Subgroups of fP groups} to construct embeddings into finitely presented groups is the concept of benign subgroups. Higman gives three equivalent definitions of which we use the first one:

\begin{Definition}
\label{DE benign subgroup}
A subgroup $H$ of a finitely generated group $G$ is called a \textit{benign subgroup} in $G$, if $G$ can be embedded in a finitely presented group $K$ which has a finitely generated subgroup $L$ such that $G \cap L = H$.
\end{Definition}

As a first evident example of a benign subgroup one may take any finitely generated subgroup $H$ in any finitely presented group $G$. Then we just have to choose $L\!=\!H$.

Let us illustrate benign subgroups by a construction giving some insight of how a benign subgroup can help to embed a group with \textit{infinitely many} relations into a finitely presented group.
Assume $A = \langle
a_0, a_1, \ldots, a_n \mathrel{|}
\text{$\big[a_0,\, h_i(a_1,\ldots,a_n)\big]=1$,\; $i=1,2,\ldots$}\,
\rangle$
is a group presentation with infinitely many relations requiring that the frist generator $a_0$ commutes with some words $h_i=h_i(a_1,\ldots,a_n)$ on the rest of generators.
Suppose the subgroup $H=\langle
h_i,\,i=1,2,\ldots \rangle$ is benign
in the free group $G=\langle
a_1, \ldots, a_n \rangle$, let $K$ be the finitely presented group containing $G$, and let
$L=\langle
l_1,\ldots,l_t
\rangle \le K$.
Then $\Gamma = K *_L t=
\langle K, t \mathrel{|}
\text{$l^t=l$ for each $l\in L$}
\rangle$ is finitely presented, since we should require  $l^t=l$ for $l_1,\ldots,l_t$ only, and since $K$ demands finitely many relations.
On the other hand,
by
Corollary~\ref{CO G*t lemma corollary} (1)
the subgroup $\Gamma' = \langle G, t \rangle$ of $\Gamma$ is equal to
$G *_{G \cap L} t = G *_{H} t$.
It remains to rename $t$ by $a_0$ to see that $G *_{H} \!t \,\cong\, A$.

\medskip

It is interesting to compare Definition~\ref{DE benign subgroup} with Theorem V (a variation of the well-known theorem on embeddings of countable groups into 2-generator groups) in earlier article \cite{HigmanNeumannNeumann}.

\medskip

The following properties have simple proofs covered in \cite{Higman Subgroups of fP groups} by Lemma~3.6 and Lemma~3.7:

\begin{Lemma}
\label{LE benign subgroups intersection and union are benign}
Let $G$ be a finitely generated group with two benign subgroups  $H,\,H'$. Then:
\begin{enumerate}
\item $H \cap H'$  is benign in $G$,

\item $\langle H, H' \rangle$ is benign in $G$.
\end{enumerate}
\end{Lemma}

\begin{Lemma}
\label{LE benign subgroup has benign image}

Let $G,\, N$ be finitely generated groups both possessing embeddings into some finitely presented groups, and let $\varphi : G \to N$ be any homomorphism. Then:
\begin{enumerate}
\item if a subgroup $H$ is benign in $G$, then its image $\varphi(H)$ is benign in $N$,

\item if a subgroup $H$ is benign in $N$, then its (complete) pre-image $\varphi^{-1}(H)$ is benign in $G$.
\end{enumerate}
\end{Lemma}

\begin{Remark}
\label{LE benign in two subgroups}
It is trivial to verify by Definition~\ref{DE benign subgroup} that the subgroup $H$ of a finitely generated group $G$ is benign in $G$, provided that $H$ is benign in a finitely generated group $G'$ containing $G$.
On the other hand, if we additionally require that $G'$ (and, therefore, also $G$) can be embedded into a finitely presented group, then
$H$ is benign in $G'$, provided that $H$ is benign in $G$. To see this just apply Lemma~\ref{LE benign subgroup has benign image} (1) to the identical embedding $G \to G'$.
Combining these facts, let us following \cite{Higman Subgroups of fP groups} agree to call $H$ \textit{benign} (without specifying in \textit{which} group $G$), whenever there is a finitely presented overgroup containing the groups discussed.
\end{Remark}

\subsection{Examples of benign subgroups}
\label{SU Basic examples of benign subgroups}

Let us collect some examples for later use. We intentionally bring them in an order displaying development of some basic ideas.
Take the free group $G=\langle a,b,c \rangle$, its subgroup
$G_0=\langle b,c \rangle$, and denote $b_i=b^{c^i}$ for $i \in \Z$ as in \ref{SU Notations for groups and homomorphisms}.

\begin{Example}
\label{EX b all powers is benign}
The subgroup $A_0= \langle b_i \mathrel{|}
i\in \Z \,\rangle$ is benign in $G_0$.
Indeed, let $\varphi$ be the isomorphism of $G_0$ sending $b,c$ to $b^c\!,c$.
Then $\Gamma\!=\! G_0\, \!*_{\varphi}\! t$ is finitely presented, and it is enough to verify
$G_0 \cap \langle b, t \rangle \!= A_0$.
Each $b_i\!=b^{c^i}\!\!=b^{t^i}$ is in $G_0 \cap \langle b, t \rangle$.
Next, applying \ref{SU The conjugates collecting process}
to any $w\in \langle b, t \rangle$
for $\X=\{b \}$
and $\Y=\{t\}$, we get $w = u \cdot v$ where $u$ is a product of factors of type $b^{\pm t^{i}}\!\!= b_i^{\pm 1}$\!, and $v=t^k$. If also $w\in G_0$, then $v=1$ by uniqueness of the normal form, i.e., $w\in A_0$.
\end{Example}

\begin{Example}
\label{EX b SOME powers is benigh}
The subgroup $A_{k,l}=\langle b_{n} \mathrel{|}
n= ik+l,\;i\in \Z\,
\rangle$ is benign in $G_0$ for any $k,l=1,2,\ldots$ To see this just apply to $A_{\,0}$ the homomorphism $\varphi$ sending $b,\,c$ to $b^{c^l}\!\!,\;c^k$.
Since $A_{k,l}=\varphi(A_0)$, it remains to use Lemma~\ref{LE benign subgroup has benign image} (1).
\end{Example}

\begin{Example}
\label{EX B_0 is benign}
The subgroup $B_0=\langle b_i \mathrel{|} i\ge 0 \rangle$ is benign in $G_0$.
Indeed, let $\varphi$ be the homomorphism  sending $b,c$ to $b^{c}\!=b_{1},\;c^2$, and let $\psi$ be the homomorphism  sending $b,c$ to $b,\,c^2$.
Denoting
$A = G_0$,\,
$B = \langle b_{1},c^2 \rangle$,\,
$C = G_0$,\,
$D = \langle b,c^2 \rangle$
we have
$B= \varphi(A)$ and
$D= \psi(C)$.
Then $\Gamma = G_0 *_{\varphi,\psi} (t,s)$ is finitely presented, and since $\langle B_0, t,s \rangle=\langle b_1, t,s \rangle$, it will be enough to show $G_0 \cap \langle B_0, t,s \rangle=B_0$. One side is evident, so we verify $G_0 \cap \langle B_0, t,s \rangle \le B_0$.

Collect some information on a transversal to $B$ in $G_0$.
For $F'\!=\!\langle b_i \mathrel{|} \text{$i$ is odd}\rangle$ and
$F''=\langle b_i \mathrel{|} \text{$i$ is even}\rangle$ we  clearly have $F' * F'' = A_0$.
Applying \ref{SU The conjugates collecting process}
for $\X=\{b\}$ and $\Y=\{c\}$, we can write each element  $g\in G_0$ as
$g=b^{\pm c^{n_1}}\!\!\cdots\, b^{\pm c^{n_s}} c^k
\;=\;b^{\pm 1}_{n_1}\cdots\, b^{\pm 1}_{n_s}\, c^k
$
for some $n_1,\ldots,n_s,\, k \in \Z$.
And $g$ is in $A_0$ if and only if $c^k=1$. Moreover, $g$ is in $F'$ if and only if all the $n_1,\ldots,n_s$ are odd.
Applying this on $g=w^{-1} w'$ with any $w,w'\in A_0$, we see that $w^{-1} w' \in B=\langle F',c^2 \rangle$ if and only if $w^{-1} w' \in F'$, i.e., $F'w=F'w'$. Thus, any transversal $T_{F'}$ to $F'$ in $A_0$ can be continued to a transversal $T_{B}$ to $B$ in whole $G_0$.
Since $A_0$ is free, $T_{F'}$ is easy to describe as follows. Write:
\begin{equation}
\label{EQ the worm of w}
w=b_{n_1}^{\varepsilon_1}\!\cdots b_{n_{r-1}}^{\varepsilon_{r-1}}\cdot
b_{n_r}^{\varepsilon_r}\!\cdots b_{n_s}^{\varepsilon_s} \in A_0
\end{equation}
where $n_r$ is the first even index, and all $n_1,\ldots,n_{r-1}$ are odd. Then $b_{n_1}^{\varepsilon_1}\!\cdots b_{n_{r-1}}^{\varepsilon_{r-1}} \in F' \le B$ and $b_{n_r}^{\varepsilon_r}\!\cdots b_{n_s}^{\varepsilon_s}\in T_{F'} \subseteq T_{B}$.
If, in addition, $w$ is from $B_0$, then all indices $n_1,\ldots,n_{s}$ are non-negative.

Similarly, any transversal $T_{F''}$ to $F''$ in $A_0$ can be continued to a transversal $T_{D}$ to $D$ in whole $G_0$. Any element from $B_0$ can be written similarly to \eqref{EQ the worm of w} such that  $n_r$ is the first odd index, all $n_1,\ldots,n_{r-1}$ are even, and all $n_1,\ldots,n_{s}$ are non-negative. Then $b_{n_1}^{\varepsilon_1}\!\cdots b_{n_{r-1}}^{\varepsilon_{r-1}} \in F'' \le D$ and $b_{n_r}^{\varepsilon_r}\!\cdots b_{n_s}^{\varepsilon_s}\in T_{F''} \subseteq T_{D}$.
Finally, set trivial transversals $T_A=T_C=\1$ because $A=C=G_0$.

Each $x\in \langle B_0, t, s \rangle$ is a product of some elements $w\in B_0$, and of some $t^{\pm 1}$\!\!,\, $s^{\pm 1}$\!.
Above preparations show how to reduce $x$ to its normal form in $\Gamma$. Scanning $x$ from right to left:

replace in $x$ a subword of type $t^{-1} w$ by $t^{-1} b_{n_1}^{\varepsilon_1}\!\cdots b_{n_s}^{\varepsilon_s} \,t \cdot t^{-1} = b_{2n_1+1}^{\varepsilon_1}\!\cdots b_{2n_s+1}^{\varepsilon_s} t^{-1}$\!\!,

replace a subword of type $s^{-1} w$ by $s^{-1} b_{n_1}^{\varepsilon_1}\!\cdots b_{n_s}^{\varepsilon_s} \,s \cdot s^{-1} = b_{2n_1}^{\varepsilon_1}\!\cdots b_{2n_s}^{\varepsilon_s} s^{-1}$\!\!,

replace a subword of type $tw$ by
$t\, b_{n_1}^{\varepsilon_1}\!\cdots b_{n_{r-1}}^{\varepsilon_{r-1}}
\, \,t^{-1} \cdot\, t \;
b_{n_r}^{\varepsilon_r}\!\cdots b_{n_s}^{\varepsilon_s}
= \varphi^{-1}(b_{n_1}^{\varepsilon_1}\!\cdots b_{n_{r-1}}^{\varepsilon_{r-1}})$
$ \cdot t \;
b_{n_r}^{\varepsilon_r}\!\cdots b_{n_s}^{\varepsilon_s}
$ $
=b_{(n_1 - 1)/2}^{\varepsilon_1}\!\cdots b_{(n_{r-1}-1)/2}^{\varepsilon_{r-1}}
\cdot
b_{n_r}^{\varepsilon_r}\!\cdots b_{n_s}^{\varepsilon_s}
$,
which is doable, as $n_1,\ldots,n_{r-1}$ are odd,

replace a subword of type $sw$ by
$s\, b_{n_1}^{\varepsilon_1}\!\cdots b_{n_{r-1}}^{\varepsilon_{r-1}}
\, \,s^{-1} \cdot \,s \;
b_{n_r}^{\varepsilon_r}\!\cdots b_{n_s}^{\varepsilon_s}
= \psi^{-1}(b_{n_1}^{\varepsilon_1}\!\cdots b_{n_{r-1}}^{\varepsilon_{r-1}})$
$ \cdot s \;
b_{n_r}^{\varepsilon_r}\!\cdots b_{n_s}^{\varepsilon_s}
$ $
=b_{n_1/2}^{\varepsilon_1}\!\cdots b_{n_{r-1}/2}^{\varepsilon_{r-1}}
\cdot
b_{n_r}^{\varepsilon_r}\!\cdots b_{n_s}^{\varepsilon_s}
$,
which is doable, as $n_1,\ldots,n_{r-1}$ are even this time.

Applying these four actions to $x$, we never get a \textit{new} $b_i^{\varepsilon_i}$ with \textit{negative} $i$, i.e.,
the normal form of $x$ consists of some factors $b_i^{\varepsilon_i}$ with non-negative $i$, and of some $t^{\pm 1}$\!\!,\; $s^{\pm 1}\!\!$.
If, moreover, $x$ is in $G_0$, then by uniqueness of the normal form we rule out such $t^{\pm 1}$\!\!,\; $s^{\pm 1}\!\!$,\, and get $g \in B_0$.
\end{Example}

\begin{comment}
The reader may compare the second half of the above proof with Lemma 3.8 in~\cite{Higman Subgroups of fP groups}. We show that $B_0$ is benign without using Lemma~3.1--Lemma~3.4 of~\cite{Higman Subgroups of fP groups}. Instead, we deduce it from the process of \ref{SU The conjugates collecting process} and from from some simple combinatorics on words.
\end{comment}

\begin{Example}
	\label{EX Uts is benign}
The subgroup $B_{k,l}=\langle b_i \mathrel{|} i <k \text{ or } i\ge l \rangle
$ is benign in $G_0$ for any fixed integers $k\le l$.
Let $\mu(x)=x^c$ be an inner automorphism on $G_0$.
Since $\mu^j(b_i)=b_i^{c^{j}}=b_{i+j}$, it is clear that $\mu^j(B_0)=\langle b_i \mathrel{|} i \ge j  \rangle$.
In particular, for $j=l$ the subgroup $\mu^l(B_0)=\langle b_i \mathrel{|} i \ge l  \rangle$ is benign in $G_0$
by previous example and by repeated application of Lemma~\ref{LE benign subgroup has benign image} (1).
$G_0$ has a homomorphism $\chi$ sending $b,c$ to $b,c^{-1}$\!.\, The image $ \chi(B_0)=\langle b_i \mathrel{|} i \le 0  \rangle$ and the image
$\mu^{k-1}\big(\chi(B_0)\big)=\langle b_i \mathrel{|} i \le k-1  \rangle$
both are benign in $G_0$.
Since $B_{k,l}$ is a free group, we have
$B_{k,l}= \big\langle
\mu^l(B_0),\; \mu^{k-1}\big(\chi(B_0)\big)
\big\rangle$, and this subgroup also is benign by
Lemma~\ref{LE benign subgroups intersection and union are benign} (2).
\end{Example}

To put some more generators into the game setup the free group $G^*=\langle
a,b,c, g,h,k \rangle$.
In analogy to $b_i$ and $b_f$ introduce $h_i=h^{k^i}$ for $i\in \Z$ and $h_f = h_0^{f(0)}\cdots h_{m-1}^{f(m-1)}$ along with $g_f=g^{h_f}$ for  $f\in \E_m$.
Using the elements $a, b_i, g, h_i$ we can choose new free subgroups of $G^*$, such as
$G_1\!=\!\langle
g,\,
h_0, \ldots, h_{m-1}
\rangle$.
Below we are going to construct some benign subgroups inside such subgroups. To make discussion simpler, we treat a group of type $G_1$ as a free group on generators $g,\,
h_0, \ldots, h_{m-1}$, ``forgetting'' the fact that $h_i$ is the conjugate $h^{k^i}$ for some $k$. In such groups we still can use the notations like $b_i$, $b_f$, $a_f$, $h_i$, $h_f$, $g_f$.
The reason of this manner of notations is to make the main proof in Section~\ref{SE The proof of Theorem} clearer.

\begin{Example}
\label{EX E_m is benign}
The subgroup
$G_{\E_m}\!=
\langle
g_f \mathrel{|} f\in \E_m
\rangle
$ is benign in  $G_1\!=\!\langle
g,\,
h_0, \ldots, h_{m-1}
\rangle$.
Indeed, for each $r=0,\ldots, m-1$ there is an isomorphism $\varphi_r$ on $G_1$ sending
$g,\,
h_0, \ldots, h_{r-1},  h_{r} , \ldots, h_{m-1}$
to
$g^{h_r}\!,\,
h_0^{h_{r}}\!, \ldots, h_{r-1}^{h_r},  h_{r} , \ldots, h_{m-1}$.
Then $\Gamma = G_1 *_{\varphi_0 ,\ldots, \varphi_{m-1}} \!(t_0 ,\ldots, t_{m-1})$ is finitely presented.
The action $g_f^{ t_r^{\pm1}}\!\!=g_{f_r^{\pm}}$ in $\Gamma$ is trivial to verify. Say, for $f=(2,5,3)$ and $r=1$ we have:
$$
\big(g^{h_0^2 \, h_1^5 \, h_2^3}\big)^{t_1} \!\!=
\big(h_2^{-3}  h_1^{-5} h_0^{-2}  \big)^{t_1} \! g^{t_1}\,
\big( h_0^2 \, h_1^5 \, h_2^3\big)^{t_1}
\!\!=
h_2^{-3}
h_1^{-5}
\big(h_0^{-2}\big)^{h_1}
g^{h_1} \,
\big(h_0^{2}\big)^{h_1}  h_1^{5}
h_2^3
=
g^{h_0^2 \, h_1^{5+1} \, h_2^3}
\!\!=
g^{h_{f_1^+}}\
$$
with $f_1^+\!\!=\!(2,6,3)$.
Thus, $L= \langle
g,\,
t_0, \ldots, t_{m-1}
\rangle$ contains  the elements $g_f=g^{t_{0}^{f(0)}\cdots \,t_{m-1}^{f(m-1)}}$ for \textit{all} $f\in \E_m$.
On the other hand,
applying \ref{SU The conjugates collecting process}
to any word $w\in L$ for
$\X=\{g\}$
and $\Y=\{t_0, \ldots, t_{m-1}\}$
we rewrite it as $w=u\cdot v$, where $u$ is a product of some conjugates of $g$ by some words in $t_i$ (and they are equal to some $g_f$ as we just saw), while $v$ is another word in $t_i$.
If $w$ also is in $G_1$, then $v=1$ by uniqueness of the normal form, and so $G_1 \cap L \subseteq G_{\E_m}$.
\end{Example}

\begin{comment}
The last two examples prove Lemma 3.9 in~\cite{Higman Subgroups of fP groups} without using Lemma~3.1--Lemma~3.4 of~\cite{Higman Subgroups of fP groups}. Instead, we deduce them from  \ref{SU The conjugates collecting process} and from from some simple  combinatorics.
\end{comment}

\begin{Example}
If $v_f=g_{f^+} \, b^{-1}_{m-1}  g_f^{-1}$, then the subgroup
$V_{\E_m}\!\!=
\langle
v_f \mathrel{|} f\in \E_m
\rangle
$ is benign in $G_2\!=\!\langle
b_{m-1},\,
g,\,
h_0, \ldots, h_{m-1}
\rangle$.
Let $\varphi_r$ be the isomorphism on $G_2$
fixing $b_{m-1}$, and with same effect on other generators as in previous example.
$\Delta\!= \! G_2 *_{\varphi_0 ,\ldots, \varphi_{m-1}} \!(t_0 ,\ldots, t_{m-1})$ is finitely presented. The equality
$v_f^{t_r^{\pm1}}\!\!=v_{f_r^\pm}^{\vphantom{1}}$ is trivial to verify in $\Delta$. \;Say, for $f=(2,5,3)$ and $r=1$:
$$
v_f^{t_1}=\!
\big(g^{h_0^2 \, h_1^5 \, h_2^{3+1}}\big)^{t_1}
(b_{m-1}^{-1})^{t_1}
\big(g^{-\,h_0^2 \, h_1^5 \, h_2^{3}}\big)^{t_1}
=
g^{h_0^2 \, h_1^{5+1} \, h_2^{3+1}}
b_{m-1}^{-1}
g^{-b_0^2 \, h_1^{5+1} \, h_2^{3}}
=
v_{f_1^+}\,.
$$
\vskip-1mm
\noindent
If $v_{(0)}\!=
g^{h_{m-1}}\, b_{m-1}^{-1}  g^{-1}
$\!\!,\;  then the above shows that
$L= \langle
v_{(0)},\, t_0 \ldots, t_{m-1}
\rangle$ contains
$v_f\!=\!
v_{(0)}^{t_{0}^{f(0)}\cdots \,t_{m-1}^{\,f(m-1)}}$\! for \textit{all} $f\in \E_m$.
On the other hand,
applying \ref{SU The conjugates collecting process}
to $w\in L$ for
$\X\!=\!\{v_{(0)}\}$
and $\Y\!=\!\{t_0, \ldots, t_{m-1}\}$ we like in previous example get
$G_2 \cap L \subseteq V_{\E_m}$. I.e., $V_{\E_m}$ is benign.
\end{Example}

\begin{Example}
\label{EX benign a b_i c_i}
If $z_f=g_f^{\vphantom8} b_f^{-1}$\!\!,\; then the subgroup $
Z_{\E_m}=
\langle \,
z_{f} \!\mathrel{|} f\in \E_m
\rangle
$ is benign in $G_3=\langle
b_0, \ldots, b_{m-1},\,
g,\,
h_0, \ldots, h_{m-1}
\rangle$.
First verify that
$\langle Z_{\E_{m-1}}, V_{\E_m} \rangle = Z_{\E_m}$.
We have $V_{\E_m}\! \le Z_{\E_m}$ because
$
v_f=g_{f^+} \, b_{m-1}^{-1} \, g_f^{- 1}
=
g_{f^+}^{\vphantom8} \, b_{f^+}^{-1} \cdot b_f^{\vphantom8} g_f^{- 1}=
z_{f^+}^{\vphantom8}\!\! \cdot z_{f}^{-1} \in Z_{\E_m}$.
\begin{comment} Say, for $f=(2,5,3)$:
$$
g^{b_0^2 \, b_1^5 \, b_2^{3+1}}\,
h_{2}^{-1}\,
a^{-b_0^2 \, b_1^5 \, b_2^{3}}
= a^{b_0^2 \, b_1^5 \, b_2^{3+1}}
\big(h_0^2 h_1^5 h_{2}^{3+1}\big)^{-1}
\!\!\cdot\,
h_0^2 h_1^5 h_{2}^{3}\;\;
a^{-b_0^2 \, b_1^5 \, b_2^{3}}
=z_{(2,5,3+1)}^{\vphantom{1}} \cdot \, z_{(2,5,3)}^{-1}.
$$
\end{comment}
And since also $Z_{\E_{m-1}}\! \le Z_{\E_{m}}$, we have $\langle Z_{\E_{m-1}}, V_{\E_{m-1}} \rangle \le Z_{\E_{m}}$. Next, show by induction on $f(m-1)$ that $z_f\in \langle Z_{\E_{m-1}}, V_{\E_{m}} \rangle $.
When $f(m-1)=0$, then $z_f\in Z_{\E_{m-1}}$.
And, since then $f^+(m-1)=1$, we from
$v_f^{\vphantom{+}}=
z_{f^+}^{\vphantom{+}}\! \cdot z_f^{-1} \in V_{\E_{m}}$  get $z_{f^+}\in \langle Z_{\E_{m-1}}, V_{\E_{m}} \rangle$.
Say, for $f=(2,5,0)$ we have
$
Z_{\E_{3-1}}\!
\ni
v_{(2,5,0)}= z_{(2,5,0+1)}^{\vphantom{+}} \cdot z_{(2,5,0)}^{-1}$, and so
$
z_{(2,5,0+1)} \in \langle Z_{\E_{3-1}}, V_{\E_{3}} \rangle
$.
The cases with $f(m\!-\!1)=1,2,\ldots$ are covered similarly. Symmetric arguments cover the case with negative $f(m\!-\!1)$. Since  $Z_{\E_{m}}=\langle Z_{\E_{m\!-\!1}}, V_{\E_{m}} \rangle$, the fact that $Z_{\E_{m}}$ is benign follows by induction: $Z_{\E_{0}} =\langle g_{(0)}^{\vphantom8} b_{(0)}^{-1} \rangle=\langle g \rangle$ is benign, as it is finitely generated.
If $Z_{\E_{m\!-\!1}}$ is benign, then $Z_{\E_{m}}=\langle Z_{\E_{m\!-\!1}}, V_{\E_{m}} \rangle$ is benign by
Lemma~\ref{LE benign subgroups intersection and union are benign} (2) and by previous example.
\end{Example}

\section{The modified proof of Theorem~\ref{TH Higman 4}}
\label{SE The proof of Theorem}

\noindent
Denote by
$\BB$
the set of all subsets $\B \subseteq \E$ for which $A_{\B}= \langle a_f \;|\; f \in \B\, \rangle$ is benign in $G=\langle a,b,c \rangle$.
First show that $\Zz,\S\in \BB$, and then verify that if a subset $\B$ is obtained from $\Zz$ and/or $\S$ by Higman operations \eqref{EQ Higman operations}, then $\B$ also is in $\BB$.
This proves Theorem~\ref{TH Higman 4} as by Theorem~3 in \cite{Higman Subgroups of fP groups} every recursively enumerable subset $\B\subseteq \mathcal E$ can be obtained that way.

\smallskip
The free group
$B\!=\!\langle a,b_i \mathrel{|} i\!\in \!\Z \rangle$  is a product $B_0*B_1$ for
$
B_0\!=\!\langle b_i \mathrel{|} i\ge 0 \rangle
$ and $
B_1=\langle a,b_i \mathrel{|}
i<0 \rangle.
$
Of these $B_0$ is benign by Example~\ref{EX B_0 is benign},
and $B_1$ is benign by
Lemma~\ref{LE benign subgroup has benign image}  and by
Lemma~\ref{LE benign subgroups intersection and union are benign} (2) because
$B_1=\big\langle \mu^{-1}\big(\chi(B_0)\big), a \big\rangle$,
where $\mu, \chi$ are those used in Example~\ref{EX Uts is benign}.
So there are finitely presented groups $K_0$ and $K_1$ which contain $G$, and which have finitely generated subgroups, respectively, $L_0$ and $L_1$ such that
$G \cap L_0 = B_0$ and
$G \cap L_1 = B_1$.
Then both HNN-extensions
$\Lambda_0 = K_0 *_{L_0} \!t$ and  $\Lambda_1 = K_1 *_{L_1} \!s$ are finitely presented.

Amalgamating $G$ in $\Lambda_0$ and $\Lambda_1$ we get the  finitely presented group $\Theta\!=\!\Lambda_0 *_{G} \Lambda_1$ in which the subgroups $G^t$ and $G^s$ intersect trivially, since
$
G^t \!\cap G^s \,\le\,
(G^t \!\cap G) \,\cap\, (G^s\! \cap G)
\,\le\, B_0 \,\cap\, B_1 = \1.
$
So in $\Theta$ we have $\langle G^t, G^s \rangle =  G^t * \,G^s$  and, thus,
$\langle B_0^t, B_1^s \rangle =  B_0^t * B_1^s \cong B_0 * B_1=B$.

Therefore, the identical automorphism on $G^t$ and the automorphism given by conjugation by $b^s$ on $G^s$ possess a common extension $\omega$ on $\langle G^t, G^s \rangle$. The respective HNN-extension
$
\Psi = \Theta *_{\omega}\! d
$
is finitely presented as $\omega$ can be defined by its values on just six generators $a^t\!\!, b^t\!\!, c^t\!\!,\; a^{\,s}\!\!, b^s\!\!, c^s$ of $G^t * G^s$.
For the generators of $B$ we in $\Psi$ clearly have:
\begin{equation}
\label{EQ action of d on F}
\text{
$b_i^d= b_i$ for $i\ge 0$,\quad
$a^d = a^b$\!\!,\;\quad
$b_i^d= b_i^b$ for $i< 0$.
}
\end{equation}

Next, take $\delta$ to be the automorphism on $G$ which sends $a,b,c$ to $a,b^c\!\!,\,c$. Then $\Delta  = \Psi  *_\delta e$  is  finitely presented.
In $\Delta$ denote $d_i = d^{e^i}$ for $i\in \Z$.
From \eqref{EQ action of d on F}, from the action of $e$ on $a,b,c$ and from $b_i^{e^j}
=
(c^{-i})^{e^j} b^{e^j} (c^{i})^{e^j}
=
c^{-i} b^{c^j} c^{i}
=b_{i+j}$ it follows:
\begin{equation}
\label{EQ action of d_i on F}
\text{
$b_i^{d_j}= b_i$ for $i\ge j$,\quad
$a^{d_j} = a^{b_j}$\!\!\!\!,\;\;\quad
$b_i^{d_j}= b_i^{b_j}$ for $i< j$.
}
\end{equation}
Here is the main feature for the sake of which $\Delta$ was built:

\begin{Lemma}
\label{LE action of d_m on f}
For any $f \in \mathcal E$ and for any fixed integer $j$
we have
$
a_f^{d_j} =\! a_{f_{j}^+}$
and
$
a_f^{\,d_j^{-1}}\!\! = a_{f_{j}^-}
$.
\end{Lemma}

\begin{proof}
Clearly, $a_f^{d_j}
\!=\! b_f^{\!-\,d_j}\,
 a^{d_j}
b_f^{\,d_j}
$\!. By \eqref{EQ action of d_i on F} the effect of conjugation of factors $b_{i}^{f(i)}$ \!in $b_f$ by $d_j$ is so that in $b_f$ all the factors standing \textit{before} $b_{j}^{f(j)}$ are conjugated by \textit{the same} $b_j$, while other factors are not changed. After cancellations of all $b_j^{\vphantom8} b_j^{-1}$ in and between the factors of $a_f^{\!-b_j}
$\!,\, $a^{b_j}$ and $
a_f^{b_j}$ we get the needed equality.
The second equality is considered similarly.
\end{proof}

\begin{Example}
\label{EX d acts on a_f}
Let $j=1$ and $f=(2,5,3)$.
Then $b_f=b_0^{2}\,b_1^5\,b_2^{3}$, and by \eqref{EQ action of d_i on F} we have:
$
a_f^{d_1}=
\big(
b_2^{\!-3}b_1^{\!-5}b_0^{\!-2}
\; a\;
b_0^{2}b_1^{5}b_2^{3} \,
\big)^{d_1}
\!\!=
b_2^{-3}\,
b_1^{-5} \,
\big(b_1^{\!-1} b_0^{\!-2} b_1^{\vphantom8}\big)
\; \big(b_1^{\!-1} a b_1^{\vphantom8}\big) \;
\big(b_1^{\!-1} b_0^{2}b_1^{\vphantom8}\big)\,
b_1^{5}
\,b_2^{3}
$,
which after cancellations is equal to
$
b_2^{\!-3}b_1^{\!-\,6}  b_0^{\!-2}
\;a\;
b_0^{2}b_1^6b_2^{3}
= a_{f_{1}^+}
$
with
$f_{1}^+ = (2,6,3)$.
\end{Example}

\begin{Lemma}
\label{LE X_Z is benign}
$\BB$ contains the sets $\Zz$ and
$\S$.
\end{Lemma}

\begin{proof}
Since  $\Zz$  consists of $f\!=\!(0)$ only, then $A_{\Zz}=\langle a \rangle$. Being finitely generated $\langle a \rangle$ is benign.

$\S$ contains the function $f=(0,1)$.
Apply Lemma~\ref{LE action of d_m on f} to $a_{f}$  repeatedly we get:
\begin{equation}
\label{EQ d0d1 acts on a_01}
a_{f}^{(d_0 d_1)^n}
\!\!=\big(a_{(0,1)}^{d_0 }\big)^{d_1  \, (d_0 d_1)^{n-1}}
\!\!\!\!=a_{(0+1,\,1)}^{d_1  \, (d_0 d_1)^{n-1}}
\!\!\!=a_{(1,\,1+1)}^{(d_0 d_1)^{n-1}}
\!\!\!=a_{(2,3)}^{(d_0 d_1)^{n-2}}
\!\!\!= \cdots
=a_{(n,n+1)}.
\end{equation}
Thus, $a_{(n,n+1)} \in \langle
a_{f}, d_0 d_1
 \rangle $ for any $n\in \Z$, and so $A_{\S}\le \langle
 a_{f}, d_0 d_1
 \rangle$.
Using \eqref{EQ elements from <x,y>}
for $x=a_{f}$
and
$y= d_0 d_1$,
we can rewrite any element $w\in \langle
a_{f}, d_0 d_1
\rangle$
as $w=u\cdot v$, where $u$ is a product of some conjugates $a_{f}^{\pm (d_0 d_1)^{n_i}}$\!\!\!\!,\, and $v=(d_0 d_1)^k$\!.
By \eqref{EQ d0d1 acts on a_01} all those conjugates are in $G$.
Thus, if also $w \in G$, then  $v\in G$.
Let us show this may happen for  $v=1$ only.
Since $d\notin G$, we can include it in any transversal $T_G$ to $G$ in $\Psi$.
Then $v = (d^{e^0} d^{e^1})^k
= (d\, e^{-1} \! d\, e)^k$ is in normal form in $\Delta$
 (with respect to $T_G$), i.e., $k=0$, if $v\in G$.
So $G \cap \langle
a_{f}, d_0 d_1
\rangle = A_{\S}$, and  $A_{\S}$ is benign.
\end{proof}

\begin{Theorem}
\label{LE closed under higman operations}
The set $\BB$ is closed under the Higman operations \eqref{EQ Higman operations}.
\end{Theorem}

This fact at once follows from Lemma~\ref{LE benign subgroups intersection and union are benign} for binary operations $\iota$ and $\upsilon$.
The proofs for unary operations
$\rho,\, \sigma,\, \tau,\, \theta,\, \zeta,\, \pi,\, \omega_m$ follow from specific cases and examples below. In each of them we suppose $\B $ is in $\BB$, and show that a Higman operation keeps it in $\BB$.

\smallskip

\textit{$\BB$ is closed under  $\rho$} because for the automorphism $\varphi$  of $G$
sending $a,b,c$ to $a,b,c^{-1}$
we have
$\varphi(b_i)=\varphi(b^{c^i})=b^{c^{-i}}=b_{-i}$. So
$A_{\rho(\B)} =  \varphi(A_{\B})$ is benign by Lemma~\ref{LE benign subgroup has benign image} (1).

\smallskip

\textit{$\BB$ is closed under  $\sigma$.}
Indeed, for the automorphism $\varphi$  of $G$
sending $a,b,c$ to $a,b^{c}\!\!,\,c$
we have
$\varphi(b_i)=\varphi(b^{c^i})=b^{c^{i+1}}\!=b_{i+1}$, and so
$A_{\sigma(\B)} =  \varphi(A_{\B})$ is benign by Lemma~\ref{LE benign subgroup has benign image} (1).

\textit{$\BB$ is closed under  $\zeta$.}
For each $f \in \B$ we
in $\Delta$
by Lemma~\ref{LE action of d_m on f} have
$a_f^{d^k}\!\!=a_{f'}^{\vphantom8}$, with
$f'(0)\!=\!f(0)+k$, and $f'(i)\!=\!f(i)$ for $i\!\neq\! 0$.
Any $f'\!\!\in\! \zeta(\B)$ can be obtained this way, and so $R\!=\!\langle\, A_{\B}, d\rangle$ contains $A_{\zeta(\B)}$.
On the other hand, for any $w \!\in\! R$ we apply  \ref{SU The conjugates collecting process}  for
$\X=\{a_f \mathrel{|} f \in \B\}$
and $\Y=\{d\}$ to get $w = u \cdot v$, where $u$ is a product of factors of type $a_f^{\pm d^{k_i}}\!\!\!\in A_{\zeta(\B)}$, and where $v=d^k$.
If also $w \in G$, then $v\!=\!1$ by uniqueness of  normal form, i.e., $w\in A_{\zeta(\B)}$ and $G \!\cap \!R= A_{\zeta(\B)}$.
Since $R$ is benign by Lemma~\ref{LE benign subgroups intersection and union are benign} (2), $ A_{\zeta(\B)}$ is benign by Lemma~\ref{LE benign subgroups intersection and union are benign} (1).

\vskip1mm

\textit{$\BB$ is closed under  $\pi$.}
In previous case take $R=\langle\, A_{\B}, d_1, d_2,\ldots\, \rangle$ instead.
$R$ contains $A_{\pi(\B)}$, and  applying \ref{SU The conjugates collecting process} for
$\X=\{a_f \mathrel{|} f \in \B \}$
and $\Y=\{d_1, d_2,\ldots\}$ we get
$G \cap \,R= A_{\pi(\B)}$.
Since $B_0$
is benign in $\langle b,c \rangle$ by Example~\ref{EX B_0 is benign}, its image  $\langle d_0, d_1,\ldots \rangle$
together with the subgroup
$\langle d_1, d_2,\ldots \rangle$
are benign in $\langle d,e \rangle$ and, thus, in $\Delta$.  Hence $R$ is benign by Lemma~\ref{LE benign subgroups intersection and union are benign} (2). Then
$A_{\pi(\B)}$ is benign by Lemma~\ref{LE benign subgroups intersection and union are benign} (1).

\vskip1mm

\textit{$\BB$ is closed under  $\theta$.}
Modify the idea of previous points. The subgroup
$\langle d_{i} \mathrel{|}
\text{all odd $i$}\, \rangle$
is benign as it is the isomorphic image of the benign subgroup $A_{2,1}$ from Example~\ref{EX b SOME powers is benigh}.
Since $A_{\B}$ is benign,  $R_1=\langle \, A_{\B},  d_{i} \mathrel{|}
\text{all odd $i$}\, \rangle$ also is benign.
Applying \ref{SU The conjugates collecting process}  for
$\X=\{a_f \mathrel{|} f \in {\B} \}$
and $\Y=\{ d_{i} \mathrel{|}
\text{all odd $i$}\, \}$ we get that
$G \cap R_1= A_{{\B}_1}$, where $f' \in {\B}_1$
if and only if there is an $f \in {\B}$ such that $f'(i)=f(i)$ for all even $i$ (there can be any values $f'(i)$ for odd $i$).
Similarity, $\langle d_{i} \mathrel{|}
\text{all even $i$}\, \rangle$
 is benign (use $A_{2,0}$ this time).
Thus, $R_2=\langle \, a,  d_{i} \mathrel{|}
\text{all even $i$}\, \rangle$ also is benign.
Applying \ref{SU The conjugates collecting process} for
$\X=\{a\}$
and $\Y=\{ d_{i} \mathrel{|}
\text{all even $i$}\, \}$ we get that
$G \cap R_2= A_{\B_2}$ where $f' \in \B_2$
if and only if $f'(i)=0$ for all odd $i$ (there can be any values $f'(i)$ for even $i$).
Then  $A_{\B_1}\! \cap A_{\B_2}$ is benign.
Since the $a_f$ are \textit{free} generators,
$A_{\B_1}\! \cap A_{\B_2}\!\!=\!A_\Y$ where $\Y\!=\!\B_1 \!\cap \B_2$.
But $\B_1 \!\cap \B_2$
clearly consists of all those functions $f'$ which coincide with some $f \in \B$ on all even $i$, and are zero elsewhere.
Let $\gamma$ be the homomorphism sending sending $a,b,c$ to $a,b,c^2$.
Then $A_{\theta(\B)}=\gamma^{-1}(A_\Y)$, and so $A_{\theta(\B)}$ is benign by Lemma~\ref{LE benign subgroup has benign image} (2).

\vskip1mm

\textit{$\BB$ is closed under  $\tau$.}
By Example~\ref{EX Uts is benign}
\, $B_{0,2}$ is benign in $G$. Take $K$ to be a finitely presented group with a finitely generated subgroup $L$ such that $G \cap L = B_{0,2}$, and set   $\Lambda = K *_{L} \! t$.
As it is easy to check
$\langle K, K^t \rangle
$ is equal to $ K *_{L} K^t$.
Since $B \cap B_{0,2}  = B_{0,2}= G^t \cap B_{0,2}$, we by
Corollary~\ref{CO G*H lemma corollary} (1)
have $\langle B, G^t \rangle = B *_{B_{\,0,2}} \!\!G^t$ in $\Lambda$.
Since $B$ is free, it has an automorphism swapping
$b_0$ and $b_1$, while fixing $a$ and $b_i$
for any $i\neq 0,1$.
This automorphism and the  identical automorphism of $G^t$ agree on $B_{0,2}$, so they have a common extension $\xi$ on entire $B *_{B_{\,0,2}} \!\!G^t$\!. Then $\Phi= \Lambda *_{\xi} q$ is finitely presented as $\xi$
can be defined by its values on just six generators $a, b_0, b_1;\; a^t\!, b^t\!, c^t$ of $B *_{B_{0,2}} \!G^t$.
For every $f\! \in \mathcal E$ we clearly have $\tau (b_f) = b_{f'}$ where
$f'(0)=f(1)$,\,
$f'(1)=f(0)$,\,
and
$f'(i)=f(i)$,\,
for $i\neq 0,1$.
Thus,  $A_{\tau(\B)}=A_{\B}^q$ in $\Phi$.
Applying Lemma~\ref{LE benign subgroup has benign image} (1) for the trivial embedding $\alpha: G \to \Phi$ we get that $A_{\B}$ is benign in $\Phi$. By the above construction and again by Lemma~\ref{LE benign subgroup has benign image} (1) $A_{\tau(\B)}=A_{\B}^{\,q}$ also is benign in $\Phi$.
The pre-image $\alpha^{-1}(A_{\tau(\B)})$ in $G$ clearly is $A_{\tau(\B)}$, which then is benign in $G$ by Lemma~\ref{LE benign subgroup has benign image} (2).

\vskip1mm

\textit{$\BB$ is closed under  $\omega_m$ for each $m=1,2,\ldots$}\;
Denote $\B_m = \B \cap \E_m = \iota(\B, \E_m)$.
Since by Example~\ref{EX E_m is benign}
\, $A_{\E_m}$ is benign, $A_{\B_m}$\! is benign also.
Clearly, $\omega_m(\B)=\omega_m(\B_m)$, and we may suppose $\B = \B_m$.
Since $B_{0,m}$ is benign
in $\langle b,c\rangle$ by Example~\ref{EX Uts is benign},  $\langle b,c\rangle$ is embeddable into some finitely presented group $K$ with a finitely generated subgroup $P$ such that
$\langle b,c\rangle \cap P = B_{0,m}$.
Set
$
\Sigma = K *_P (g,h,k)
$, where each of $g,h,k$ stabilizes $P$\, (and also $B_{0,m}$).

Since the intersection of $\langle b_0,\ldots, b_{m-1} \rangle$ with $B_{0,m}$ and, thus, with $P$ is trivial, by  Lemma~\ref{LE subgroups in HNN-extension} or directly by the normal form theorem the elements $b_0,\ldots, b_{m-1},\, g,h,k $ generate in $\Sigma$ the free product $\langle b_0,\ldots, b_{m-1}\rangle * \langle g,h,k \rangle$, i.e., the free subgroup $\langle b_0,\ldots, b_{m-1},\, g,h,k\rangle$.
Denoting $h_i=h^{k^i}$ for $i \in \Z$,
we get another free subgroup
$\langle b_0,\ldots, b_{m-1},\, g,\, h_0,\ldots, h_{m-1} \rangle$ in $\Sigma$.
Setting $
h_f=h_0^{f(0)} \!\cdots\, h_{m-1}^{f(m-1)}
$ and $g_f=g^{h_f}$ we
by Example~\ref{EX benign a b_i c_i} get its benign subgroup
$L=\langle g_f^{\vphantom8} b_f^{-1}
\mathrel{|}
f\in \mathcal E_m \rangle$. I.e., $\Sigma$ is embeddable into a finitely presented  $M$ with a finitely generated subgroup $R$ such that
$\Sigma \cap R = L$. Take $\Pi=M *_{R} a$,
with some $a$ fixing $R$.

If we show that $a,b,c$ generate a free subgroup in $\Pi$, we can identify it with our initial group $G=\langle a,b,c \rangle$.
By Lemma~\ref{LE subgroups in HNN-extension} that will be shown if we verify
$\langle a \rangle \cap R=\1$ (which is evident) and $\langle b, c \rangle \cap R=\1$.  By construction $\langle b, c \rangle \cap R=\langle b, c \rangle \cap L$.
Any two distinct words on letters $b_0,\ldots, b_{m-1}$ are distinct modulo $B_{0,m}$ and, thus, are distinct in $K$ modulo $P$. I.e., we can choose a transversal $T_P$ to $P$ in $K$ containing all the words on $b_0,\ldots, b_{m-1}$. Any non-trivial product $v$ of elements of type $g_f^{\vphantom8} b_f^{-1}\!=\! g^{h_0^{f(0)} \!\cdots\, h_{m-1}^{f(m-1)}} b_{m-1}^{-f(m-1)} \!\cdots b_{0}^{-f(0)}$ already is in normal form in $\Sigma$ (with respect to $T_P$) because it is a product of stable letters and transversal's elements. It will be in $\langle b,c \rangle$ only if no stable letter is preset, i.e., if it is trivial.

Since  $G=\langle a,b,c \rangle$ is free, using its isomorphism $\rho$ sending
$a, b, c$ to $a, b^{c^m}\!\!\!\!\!,\;c$ define
$\Omega=\Pi *_{\rho} r$.
Denote $W_{\B}=\langle g_f\!,\, a,\, r \mathrel{|} f\in \B\rangle$. For any  $l\in \omega_m \B$ the element $a_l=a^{b_l}$ is in $W_{\B}$. Let us show this simple fact by a step-by-step construction example.
Let $m=3$ and let  $(2,5,3)$, $(7,2,4) \in \B$.
Then $\omega_3 \B$ contains, say,
$
l=(
0,0,0,\;
7,2,4,\;
0,0,0,\;
2,5,3,\;
7,2,4)
$. To show that $a^{b_l}\in W_{\B}$ start by the initial function $l_1=(7,2,4)$, and then modify it to achieve the value $l$ above.
The relation $(g_f^{\vphantom8} b_f^{-1})^a=g_f^{\vphantom8} b_f^{-1}$  is equivalent to $a^{g_f\vphantom{j}}\!=a^{b_f}$.

\vskip-1mm
\textit{Step 1.} Since  $f=l_1=(7,2,4)$ is in $ \B$, then $g_{l_1} \in W_{\B}$, and so
$
a^{g_{l_1}}\!\!=
a^{b_{l_1}}=
a^{b_0^{7}\,b_1^{2}b_2^{4}}
\in W_{\B}$.

\textit{Step 2.}  Conjugating the above element by $r$ we get that
$$
(a^{b_{l_1}})^r\!\!=
(a^r)^{(b_0^{7}\,b_1^{2}b_2^{4})^r}\!\!=
a^{b_3^{7}\,b_4^{2}b_5^{4}}
=
a^{b_0^{0}\,b_1^{0}b_2^{0} \;\cdot\; b_3^{7}\,b_4^{2}b_5^{4}}=
a^{b_{l_2}}\in W_{\B}
\text{\quad (with $l_2=(0,0,0,\;
7,2,4)$)}.
$$

Conjugate the above by
$g_f$ for $f=(2,5,3)$. We have:
$
\big(\! a^{b_{l_2}}\big)^{g_f}
\!\!\!=\,
a^{b_{l_2}\cdot \,g_f}
\!\!= a^{b_3^{7}\,b_4^{2}b_5^{4}\;\cdot \,g_f}\!
$.

\textit{Step 3.} Each of $g,h,k$ commutes with each of
$b_i$ for $i <0$ or $i\ge m=3$,
and so $g_f$ commutes with $b_3^{7}\,b_4^{2}b_5^{4}$, and
$
a^{b_{l_2}\cdot \,g_f}
=
a^{g_f \cdot \, b_3^{7}\,b_4^{2}b_5^{4}}
$.
Then again applying step 1 we get:
\vskip-3mm
$$
\big(\!a^{g_f}\!\big)^{b_3^{7}\,b_4^{2}b_5^{4}}
=
a^{b_0^{2}\,b_1^{5}b_2^{3} \;\cdot\; b_3^{7}\,b_4^{2}b_5^{4}}=
a^{b_{l_3}}\in W_{\B}
\text{\quad (with $l_3=(2,5,3,\;
7,2,4)$)}.
$$
Then we apply step 2 \textit{twice}, i.e., conjugate the above by $r^2$ to get
the element $a^{b_{l_4}}$ with
$
l_4=(
0,0,0,\;
0,0,0,\;
2,5,3,\;
7,2,4)
$.
Next apply step 3 and step 1 again to conjugate
$a^{b_{l_4}}$ by
$g_f$ for $f=(7,2,4)$. We get the element
$a^{b_{l_5}}$ with
$
l_5=(
7,2,4,\;
0,0,0,\;
2,5,3,\;
7,2,4)
$.
Finally, we again apply step 2, i.e., conjugate $a^{b_{l_5}}$  by $r$ to discover in $ W_{\B}$ the element $a^{b_{l}}=a_l$.
Since this procedure can easily be performed for an arbitrary $l \in \omega_m \B$, we get that $A_{\omega_m \B} \le W_{\B}$.

\vskip1mm
Let us single out an auxiliary ``slimmer'' subgroup $\Omega'$ in $\Omega$.
Since $\langle b,c\rangle \cap P = B_{0,m}$,
the subgroup $\big\langle \langle b,c\rangle, g,h,k \big\rangle$ is equal to $\Sigma' = \langle b,c\rangle *_{B_{0,m}} (g,h,k)$.
Since $\Sigma'$ contains each of $b,c, g,h,k$ it also contains $L$. Thus, $\Sigma' \cap R = L$, and using Lemma~\ref{LE subgroups in HNN-extension} we see that  $\langle \Sigma', a \rangle$ is equal to $\Pi'= \Sigma' *_L a$.
Finally, since $a,b,c\in \Pi'$, the subgroup $ \langle \Pi', r \rangle$ is equal to $\Omega' = \Pi' *_\rho r$.
The advantage of $\Omega'$ is that it is an extension of $\langle b,c\rangle$ by means of three ``nested'' HNN-extensions, and we, thus, possess the \textit{full} list of defining relations of $\Omega'$.

Since any $w \in W_{\B}$ also is in $\Omega'$, it can be brought to its normal form evolving $r$ and some elements from $\Pi'$.
The latters can in turn be brought to normal forms evolving $a$ and some elements from $\Sigma'$.
Then the latters can further be brought to normal forms evolving $g, h,k$ and some elements from $\langle b,c\rangle$.
I.e., $w$ can be brought to a \textit{unique ``nested'' normal form}.
Detect the cases when it evolves $a,b,c$ only.
The only relations of $\Omega'$ evolving $g,h,k$ are  $a^{g_f}\!=a^{b_f}$. Thus, the only way by which $g,h,k$ may be eliminated in the normal form  is to have in $w$ subwords of type $g_f^{\!-1} a\, g_f^{\vphantom8} = a^{g_f}$\!,\; to replace them by respective $a^{b_f}\in\langle a,b,c\rangle$.
If after that some subwords $g_f$ still remain, then three scenario cases are possible:

\textit{Case 1}. We may have a subword of type $w'=g_f^{-1}  a^{b_l}  g_f^{\vphantom8}$ for
such an $l$ that $l(i)=0$ for $i=0,\ldots,m-1$.
Check example of step 1, when this is achieved for $l=l_2=(0,0,0,\;
7,2,4)$ and $f=(2,5,3)$.
Then just replace $w'$ by $a^{b_{l_3}}$ for an $l_3\in \omega_m \B$ (e.g. $l_3=(2,5,3,\;
7,2,4)$).

\textit{Case 2}. If  $w'=g_f^{-1}  a^{b_l}  g_f$,\, but the condition $l(i)=0$ fails for an $i=0,\ldots,m-1$, then $g_f$ does \textit{not} commute with $b_l$, so we cannot apply the relation $a^{g_f}\!=a^{b_f}$, and so $w \notin \langle a,b,c\rangle$.
Turning to  example in steps 1--3, notice that for, say, $f=(2,5,3) \in \B$ we may \textit{never} get something like
$a^{(g_f)^{\,2} }\!\!
=(a^{b_0^{2}\,b_1^{5}\,b_2^{3}})^{g_f}\!\!
=a^{(b_0^{2}\,b_1^{5}\,b_2^{3})^2}
$ because $g_f$ does not commute with $b_0,  b_1,  b_2$. That is, all the \textit{new} functions $l$ we get are from $\omega_m \B$ \textit{only}.

\textit{Case 3}. If $g_f$ is in $w$, but not in a subword $g_f^{-1}  a^{b_l}  g_f$, we again have $w \notin \langle a,b,c\rangle$.

This means, if $w \in \langle a,b,c\rangle$, then  elimination of  $g,h,k$  turns $w$ to a product of elements from  $\langle r\rangle$ and of some  $a^{b_l}$ for some $l\in \omega_m \B$ ($a$ also is of that type, as $(0)\in\B$).
Now apply  \ref{SU The conjugates collecting process}  for
$\X=\{a^{b_l} \mathrel{|} l\in \omega_m \B \}$
and $\Y=\{r\}$ to state that
$w$ is a product of some power $r^k$ and of some elements each of which is an $a^{b_l}$ conjugated by a power $r^{n_i}$ of $r$.
These conjugates are in $\omega_m \B$, and so $w\in \langle a,b,c\rangle$ if and only if $k=0$, i.e., if
$w\in A_{\omega_m \B}$.

We have now done all the hard work, and it remains to notice that since $G=\langle a,b,c\rangle$ is a free group,  $G \cap W_{\B}=A_{\omega_m \B}$.
Since $A_{\B}$ is benign, its image
$\langle g_f \mathrel{|} f\in \B\rangle$ also is benign. Then
$W_{\B}=\langle g_f\!,\, a,\, r \mathrel{|} f\in \B\rangle$ is benign by Lemma~\ref{LE benign subgroups intersection and union are benign} (2), and  $A_{\omega_m \B}$ is benign by Lemma~\ref{LE benign subgroups intersection and union are benign} (1).

redpajama_set_name: RedPajamaArXiv


text:
Home :: Military :: Library :: News :: 2004 :: May ::
Military Menu
SLUG: 2-315765 Sri Lanka Peace Talks (L-O)
NOTE NUMBER:
TYPE=CORRESPONDENT REPORT
NUMBER=2-315765
TITLE=SRI LANKA/PEACE TALKS (L-ONLY)
BYLINE=ANJANA PASRICHA
DATELINE=NEW DELHI
VOICED AT:
INTRO: Norwegian Foreign Minister Jan Petersen (PRONO: yahn PEHTER-sen) is visiting Sri Lanka to push efforts to restart a peace process between the government and Tamil rebels. As Anjana Pasricha reports from New Delhi, the Sri Lankan government says it remains committed to ending the island's long ethnic conflict.
TEXT: A presidential spokesman says Sri Lanka's commitment to holding peace talks is strong.
Norwegian Foreign Minister Jan Petersen arrived to meet with both President Chandrika Kumaratunga and the Tamil rebels. He hopes to find ways to kick-start negotiations that have been stalled for more than a year.
Norwegian mediators left Sri Lanka six-months ago, after political rivalry between the president and prime minister made it difficult to continue their efforts. Last month, the president's party won parliamentary elections, ousted the prime minister, and invited the Norwegians back.
Mr. Petersen's delegation is the second Norwegian team to visit Sri Lanka in the past two weeks.
Rohan Edresinghe is a political analyst with Colombo's Center for Policy Alternatives. He says that despite the efforts to restart negotiations, concern remains that the president's hard-line allies could prove to be a stumbling block in establishing peace.
/// EDRESINGHE ACT ///
Some of the major actors in the international community are doing their best to get the peace process to start again, but I think there are still enormous challenges within the government that have to be sorted out before these talks can start.
/// END ACT ///
The new government is trying to address some of those challenges. A day before Mr. Petersen's visit, it agreed to recognize the Tamil Tigers as the sole representatives of the island's Tamil minority.
The government said it was doing this in response to "political and ground" realities. It was a reversal for the president's party, which had earlier said it wanted talks to include the Tamil Tigers and other Tamil parties.
The Norwegian mediators also are expected to discuss a series of minor clashes in the past two weeks that have led to the deaths of 12 people, including a government soldier and several rebels. The Tamil Tigers say government troops were involved in the deaths of the rebels, and warn this could jeopardize the peace process.
Although peace talks are deadlocked, the two-year ceasefire mediated by Norway has mostly held.
The Tamil Tigers fought for two decades for a separate homeland for the minority Tamil community, but have agreed to give up their struggle in exchange for wide-ranging autonomy in the Tamil-dominated north and east. (SIGNED)
NEB/HK/AP/KPD/RAE

redpajama_set_name: RedPajamaCommonCrawl


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£32569 to £43247 per annum (pro-rata if applicable) depending on skills and experience.
This post Clinical Research Fellow (CRF) is fully funded by Nottingham University Hospitals Charity (supported by the Stanley and Marion Irish Nee Bethell Fellowship). This Clinical Research Fellowship will involve undertaking a laboratory project investigating factors that underpin the sensitivity of multiple myeloma to drugs. The Research Fellow will register for a higher research degree (MD) for which the University's fees will be supported by the Fellowship.
The CRF will be at, or above, Specialty Trainee level 3 (ST3) in Haematology and will have previously attained successful outcomes in Annual Assessments of Competency Progression (ARCPs) or equivalent. This post does not attract a National Training Number (NTN) but existing NTN holders in Haematology may apply to their host clinical training programme for time out of programme for research (OOPR). Further information on the OOPR process is available from an NTN holder's clinical training programme directors in the region in which the NTN is currently held.
The successful candidate will work as part of a multi-professional team including scientists and clinicians, working under the direction of Dr Claire Seedhouse, Associate Professor at the University of Nottingham and Dr Dean Smith Consultant Haematologist within the Clinical Haematology department of Nottingham University Hospitals NHS Trust.
The Haematology Research Group encompasses both clinical and laboratory research. Our aim is to learn more about learn more about the biology of the diseases, characterise new drug targets, and identify better treatments for people with blood-related disorders.
Our laboratory research focuses on understanding both the pathogenesis of blood cancers and the chemoresistance phenotype often seen in the diseases. Research aims to identify ways to maximise therapeutic efficacy of existing drugs and evaluate new treatments. The CRF will explore the apoptotic response to drugs in multiple myeloma. All MD students have access to the School of Medicine's Doctoral Researcher Development Programme which offers a broad range of research-based and transferable skills courses.
The Myeloma service is led by Dr Dean Smith and includes 4 consultants and 2 Myeloma Clinical Nurse Specialists. A multidisciplinary myeloma clinic is held twice a week. In addition, there are 3 nurse-led clinics weekly and an MGUS clinic monthly. Approximately 120 new patients are seen every year with 800 in active follow-up.
The person appointed will have dedicated time to develop clinical competencies through training in the management of patients with myeloma within this department. An Honorary contract will be sought from Nottingham University Hospitals NHS Trust.
This is a full time, 38.50h per week, or part time (0.6FTE), will be offered on a fixed term contract from 01 August 2019 until 31 July 2021.


Eschenbach offers a wide range of hand-held magnifiers from which to choose. We carry illuminated and non-illuminated hand-held magnifiers with a variety of lens designs – including aspheric, aplanatic, biconvex, and the hi-tech diffractive lens. There are also options of available light sources to consider when choosing a hand-held magnifier. These include incandescent lighting or light emitting diodes (LEDs).


text:
#ifndef __BehaviorDribble_H__
#define __BehaviorDribble_H__

#include "BehaviorBase.h"
#include "VisualSystem.h"
#include <vector>

class BehaviorDribbleExecuter
    : public BehaviorExecuterBase<BehaviorAttackData> {
public:
  BehaviorDribbleExecuter(Agent &agent);
  virtual ~BehaviorDribbleExecuter(void);

  bool Execute(const ActiveBehavior &);

  void SubmitVisualRequest(const ActiveBehavior &dribble, double plus = 0.0) {
    Assert(dribble.mKeyOppGB.mUnum >= 0);
    Assert(dribble.mKeyOppGT.mUnum >= 0);

    VisualSystem::instance().RaisePlayer(mAgent, -dribble.mKeyOppGB.mUnum,
                                         2.0 + plus);
    VisualSystem::instance().RaisePlayer(mAgent, -dribble.mKeyOppGT.mUnum,
                                         2.0 + plus);
  }

  static const BehaviorType BEHAVIOR_TYPE;
};

class BehaviorDribblePlanner : public BehaviorPlannerBase<BehaviorAttackData> {
public:
  BehaviorDribblePlanner(Agent &agent);
  virtual ~BehaviorDribblePlanner(void);

  void Plan(std::list<ActiveBehavior> &behavior_list);
};

#endif

redpajama_set_name: RedPajamaGithub


text:
Wolfgang Haeseler (* 23. August 1929; † 3. Mai 2008 in Kassel) war ein deutscher Architekt.

Leben
Wolfgang Haeseler absolvierte sein Architekturstudium von 1948 bis 1951 an der Staatsbauschule Kassel und anschließend von 1952 bis 1956 an der Technischen Hochschule Darmstadt. Ab 1960 führte er in Kassel ein Büro gemeinsam Kurt von Wild, um es nach dessen Tod selbständig weiterzuführen. Ein Schwerpunkt seiner Arbeit lag im Bau von Kirchen und Krankenhäusern. Haeseler engagierte sich u. a. im BDA Hessen, im Kasseler Kunstverein und im Kunstbeirat der Stadt Kassel.

Werk (Auswahl, teils mit Kurt von Wild)
 Kassel, Kirche am Kurhessischen Diakonissenhaus (Mutterhauskirche), 1962, Turm 1983
 Bad Wildungen, Friedenskirche, 1962
 Marburg-Ockershausen, Matthäuskirche, 1963
 Kassel, Jakobuskirche, 1966
 Kassel, Brandkasse, 1973
 Kassel, Anbau an  Heinrich-Schütz-Schule (Ursprungsbau von Heinrich Tessenow: 1930, Anbau von Haeseler: 1974/75)

Literatur
 Berthold Hinz, Andreas Tacke (Hrsg.): Architekturführer. Architectural Guide Kassel, Berlin 2002, S. 41, 71.
 Karin Berkemann: "Es kommt drauf an, was man draus macht." Raumwahrnehmungen in Beton-Kirchen der Nach-kriegszeit. In: kunst und kirche, 66, 2003, S. 154–158.

Weblinks
 Traueranzeigen für Wolfgang Haeseler. trauer.hna.de; abgerufen am 26. Dezember 2020.
 ylvia Stöbe: SWolfgang Haeseler (23.8.1929 – 3.5.2008). (PDF; 158 kB) architektursalon-kassel.de; Nachruf; abgerufen am 26. Dezember 2020.
 Geschichte des Hauses.  Agaplesion Haus Salem; abgerufen am 26. Dezember 2020.

Architekt (Kassel)
Deutscher
Geboren 1929
Gestorben 2008
Mann

redpajama_set_name: RedPajamaWikipedia


text:
PM Narendra Modi welcomes Canadian PM Justin Trudeau
Last updated on - Feb 19, 2018
Let's take a look at Canadian Prime Minister Justin Trudeau's India visit...
Let's take a look at Canadian Prime Minister Justin Trudeau's India visit...Read Less
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Canadian Prime Minister Justin Trudeau and his family during their visit at the Golden Temple in Amritsar - Photogallery
Canadian Prime Minister Justin Trudeau and his family during their visit at the Golden Temple in Amritsar. (PTI)
See more of : Justin Trudeau
Canadian Prime Minister Justin Trudeau and his family during their visit at the Golden Temple in Amritsar. (Photo: Instagram) (BCCL)
Canadian Prime Minister Justin Trudeau during his visit at the Golden Temple in Amritsar - Photogallery
Canadian Prime Minister Justin Trudeau during his visit at the Golden Temple in Amritsar. (Photo: Instagram)
Canadian Prime Minister Justin Trudeau and his family during their visit at the Golden Temple in Amritsar. (Photo: Instagram)
Canadian Prime Minister Justin Trudeau being received by Union Minister of State (Independent Charge) for Housing and Urban Affairs Hardeep Singh Puri and Cultural Affairs & Tourism Minister Navjot Singh Sidhu on his arrival in Amritsar - Photogallery
Canadian Prime Minister Justin Trudeau being received by Union Minister of State (Independent Charge) for Housing and Urban Affairs Hardeep Singh Puri and Cultural Affairs & Tourism Minister Navjot Singh Sidhu on his arrival in Amritsar. (PTI)
See more of : Navjot singh sidhu, Justin Trudeau, Hardeep Singh Puri
Navjot singh sidhu, Justin Trudeau, Hardeep Singh Puri
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Prime Minister Justin Trudeau, his wife Sophie Gregoire Trudeau, their daughter Ella Grace and son Xavier pose with Bollywood actor Shah Rukh Khan in Mumbai - Photogallery
Prime Minister Justin Trudeau, his wife Sophie Gregoire Trudeau, their daughter Ella Grace and son Xavier pose with Bollywood actor Shah Rukh Khan in Mumbai. (PTI)
See more of : Shah Rukh Khan, Justin Trudeau
Shah Rukh Khan, Justin Trudeau
/news/india/justin-trudeau/aamir-khan/eventshow/63009203.cms
Aamir Khan poses with Canadian Prime Minister Justin Trudeau during an event held in Mumbai - Photogallery
Aamir Khan poses with Canadian Prime Minister Justin Trudeau during an event held in Mumbai. (PTI)
See more of : Aamir Khan
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Canadian Prime Minister Justin Trudeau and his family pose for a photograph during their visit at Akshardham Temple in Gujarat - Photogallery
Canadian Prime Minister Justin Trudeau and his family pose for a photograph during their visit at Akshardham Temple in Gujarat. (Photo: Ashwin Sadhu) (BCCL)
Canadian Prime Minister Justin Trudeau and his family arrive at Akshardham Temple in Gujarat - Photogallery
Canadian Prime Minister Justin Trudeau and his family arrive at Akshardham Temple in Gujarat. (Photo: Ashwin Sadhu) (BCCL)
Canadian Prime Minister Justin Trudeau and his family during their visit at Akshardham Temple in Gujarat - Photogallery
Canadian Prime Minister Justin Trudeau and his family during their visit at Akshardham Temple in Gujarat. (Photo: Ashwin Sadhu) (BCCL)


DEIMOS Engenharia is a private Portuguese company founded in 2002, operating in aerospace activities, and is part of the Elecnor Group. Elecnor is a large Spanish group specialised in integrated management and promotion of projects and infrastructure development.
Being also a member of the Elecnor Deimos Group DEIMOS Space has a relevant and applicable background and recognised experience in Galileo and GNSS Technology in general, and in the signal processing techniques and navigation algorithms in particular.
GeoNumerics (Barcelona, Spain, since 2002) is a small team with a long experience dedicated to geomatics. Its mission is to develop state-of-the-art geomatic technology.
TECNALIA is the first leading private and independent research and technology organisation in Spain and the 5th largest in Europe, employing 1445 expert and international people (164 PhDs), income of 118M€ and over 4.000 clients.
The Nottingham Geospatial Institute (NGI) is part of the Faculty of Engineering at the University of Nottingham. The Faculty has 2,800 undergraduate and 1,000 postgraduate students in the UK. It has approximately 375 research active staff and incorporates many world-class Research Groups, Centres and Institutes.
The Max-Planck-Institute for Biogeochemistry (MPI-BGC) is a research institute of the German Max-Planck Society (MAX), founded in 1997.
GAMMA Remote Sensing Research and Consulting AG (GAMMA) is a Swiss corporation (Aktiengesellschaft - AG).
Both its teaching and research activities make DICII one of the most active departments of Tor Vergata University of Rome. Its permanent staff presently consists of 72 full, associate and assistant professors, and the total number of active people exceeds 130.
UNESP was created in 1976. It has consolidated a project joining scientific, technological, economic, cultural and social development committed to democratic principles and clearly conscious of its status of a governmental institution, inserted and highly active in Brazilian society.
MundoGEO Publishing was created in 1998 with the mission of being the convergence point of the geomatics community and location services, trading experiences, professional updating and business generation.


James A. Devlin, age 90 years, of Easton PA., died Monday, April 1, 2019, at St. Luke's Hospital, Anderson Campus, Bethlehem Township, PA.
Born in Hopewell, NJ, November 17, 1928, son of John Joseph Devlin and Sarah Cannaughton Devlin, Jim had formerly resided in Flemington, NJ for several years and had resided in Easton, PA since 2002.
Jim was a United States Air Force Veteran of the Korean War serving in Alaska. He was an office administrator with Public Service Electric & Gas where he was employed for thirty-two years before retiring. Jim was also a member of the Hopewell, NJ American Legion Post.
Surviving are his wife, Mary Ann Hindak Devlin; five sons, James A. Devlin, Jr. (Peggi Ann Anderson) of West Trenton, NJ, Jeffrey M. Devlin (Stacy) of Hamilton, NJ, Brian P. Devlin (Linda) of Wirtz, VA, Daniel T. Devlin of Hopewell, NJ and Christopher S. Devlin (Candy) of Yardley, PA; nine grandchildren and one great grandchild. Also surviving are three step-children, June Cimino and Susan Kertesz, both of Flemington, NJ and Paul Lazorisak of Hillsborough, NJ.
Funeral services with military honors will be held Wednesday, April 10, 2019, at 1:30 PM in the Holcombe-Fisher Funeral Home, 147 Main Street, Flemington, NJ. Calling hours will be Wednesday April 10, 2019, from 1:00 PM -1:30PM at the funeral home. Interment will be private.
"Email Address" would like to share the life celebration of James A. Devlin. Click on the "link" to go to share a favorite memory or leave a condolence message for the family.


text:
Reimagining Food Allergy Management
219 Design has deep knowledge and experience developing production-ready smart devices. When Allergy Amulet needed help productizing the core technology - 219 Design was a perfect choice. This collaborative effort has resulted in an accurate, affordable, wearable, detection device that significantly improves the standard of care for those managing food allergies.
To take the worry off your plate
About Allergy Amulet
Food allergies are a rapidly growing epidemic with no cure in sight. Even trace amounts of a food allergen can be fatal. Allergy Amulet has developed the world's smallest and fastest first-of-its-kind consumer food allergen sensor by pairing molecularly imprinted polymer (MIP) technology with an electrochemical system to detect target allergenic ingredients. In seconds, users will be able to detect the presence or absence of the top eight most common food allergens (peanuts, tree nuts, fish, shellfish, wheat, eggs, milk, and soy), as well as gluten and sesame. Its sleek and portable design makes its proprietary sensor technology easy to bring everywhere you go on a keychain, necklace, or wristband.
"219 Design's electronic and firmware experts packed our tiny wearable allergen sensor full of functionality. They worked seamlessly with two other firms to bring our vision to life, and we are thrilled with the results."
Abigail Barnes, Co-Founder & CEO, Allergy Amulet
Teamwork Makes the Dream Work
Allergy Amulet wanted a high level of functionality in a small and beautiful package. The project requirements provided an opportunity for 219 Design to work closely with two other partners to identify different ways to leverage potentiostat technology, miniaturize the design, and find the right balance of technical functionality and usability requirements. The result is a beautiful, portable, connected, affordable, and easy-to-use device that can be worn or carried in a pocket or pack.
Optimizing Functionality
The 219 Design team focused on solving challenges around miniaturization, wearable battery power, and Bluetooth connectivity. We developed the electronics and firmware that make up the Amulet portion of their detection system. The reader houses the electrochemical system, as well as an OLED display for simple user interaction and clear result readout. The reader is fully functional as a standalone device and further includes Bluetooth low energy (BLE) connectivity, enabling enhanced user interaction and firmware updates when paired with the Allergy Amulet app. The design is optimized for size and power and can run for days on a single battery charge.
Many Early Successes
The result is a sleek design that brings the ability to detect food allergens to the masses through a wearable that users can seamlessly integrate into their daily lives. The detection platform leverages the latest technology and materials to provide fast and accurate consumer testing. The breakthrough and patented sensor technology tests are positioned to offer affordability, speed, and a superior user experience for this new product category.
Allergy Amulet pre-orders were wildly successful, selling out in approximately four hours. Allergy Amulet is also a female founded and led company that is working hard to overcome the gender investment gap, having raised roughly five million in seed funding so far.
"I'm proud of 219 Design's role in developing the Allergy Amulet. It has been a pleasure working with the Allergy Amulet team and its partners to develop a product with the potential to change the lives for the millions affected by life-threatening food allergies."
Alex Wood, Partner, 219 Design
We are excited to see Allergy Amulet continue to succeed and grow. 219 will continue to support their launch in the coming months and we look forward to helping them develop each new generation as technology continues to advance and evolve. In the meantime, we are ready to collaborate on the next game-changing product.


Is Chefs Catering Halal certified?
Chefs catering is officially an Halal Certified Caterer on 3rd February 2017 and under the entity of Wonder Rice Pte. Ltd.
Standard buffet catering set up does not include service staff. The hiring of service staff is chargeable at S$80 per staff for 4 hours.