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Volumes of Discrete Groups and Topological Complexity of Homology Spheres
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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?
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Finite Index Rigidity of Relatively Hyperbolic Groups
Torsion-free relatively hyperbolic groups with non-relatively-hyperbolic peripheral subgroups are finite index rigid, and peripheral-structure-preserving isomorphisms between finite index subgroups force equal indices.
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