  Let $M$ be any compact four-dimensional PL-manifold with or without boundary
(e.g. the four-dimensional sphere or ball). Consider the space $T(M)$ of all
simplicial isomorphism classes of triangulations of $M$ endowed with the metric
defined as the minimal number of bistellar transformations required to
transform one of two considered triangulations into the other. Our main result
is the existence of an absolute constant $C>1$ such that for every $m$ and all
sufficiently large $N$ there exist more than $C^N$ triangulations of $M$ with
at most $N$ simplices such that pairwise distances between them are greater
than $2^{2^{\ldots^{2^N}}}$ ($m$ times).
  This result follows from a similar result for the space of all balanced
presentations of the trivial group. ("Balanced" means that the number of
generators equals to the number of relations). This space is endowed with the
metric defined as the minimal number of Tietze transformations between finite
presentations. We prove a similar exponential lower bound for the number of
balanced presentations of length $\leq N$ with four generators that are
pairwise $2^{2^{\ldots^{2^N}}}$-far from each other. If one does not fix the
number of generators, then we establish a super-exponential lower bound
$N^{const\ N}$ for the number of balanced presentations of length $\leq N$ that
are $2^{2^{\ldots^{2^N}}}$-far from each other.

Geometric Topology