Sizes of spaces of triangulations of 4-manifolds and balanced presentations of the trivial group

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.