Volumes of Discrete Groups and Topological Complexity of Homology Spheres

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 $M\ge N\iff Mor(M,N)\ne \varphi$. 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-Poincaré characteristic. In particular, what influence does the deficiency have on the structure of an infinite group?

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