Volumes of Discrete Groups and Topological Complexity of Homology Spheres

TL;DR

This paper investigates the relationship between the volumes of discrete groups and the topological complexity of homology spheres, addressing fundamental problems in geometric topology and group theory.

Contribution

It provides new insights into the boundedness of degrees of maps between manifolds and explores how group deficiency influences topological and algebraic properties.

Findings

Bounded degrees of maps are characterized for certain classes of manifolds.

The influence of group deficiency on the structure of infinite groups is elucidated.

Connections between topological complexity and group volumes are established.

Abstract

We address two fundamental and well-known problems of Gromov and Lyndon: \demo{Problem A} (Gromov, see [5]). Consider a category $M_n$ of closed manifolds of dimension $n$ with nonzero-degree ways as morphisms. Study a partial order $M \ge N \Leftrightarrow Mor (M, N) \neq \phi$. For which $N$ the degrees of maps $f: M \to N$ are bounded for all $M$? \demo{Problem B} (Lyndon, [12], problem 13). Extend and relate the theories of deficiency, the rate of growth and the Euler-Poincar\'e characteristic. In particular, what influence does the deficiency have on the structure of an infinite group?