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Homology and homotopy complexity in negative curvature. (English) Zbl 1448.57032

The main theorem of the paper states that for a negatively curved manifold one can bound the size of the torsion part in homology using the volume of the manifold. More precisely, given a complete negatively curved manifold \(M\) of dimension \(d \neq 3\) there exists a constant \(C>0\) such that \[\text{log}|\text{tors}(H_k(M;\mathbb{Z}))| < C \cdot\text{Vol}(M) , \] where on the left we consider the logarithm of the cardinality of the torsion part.
To prove the statement the authors find an effective thick-thin decomposition of the manifold. Indeed they show that there exists a submanifold of \(M\) which is homotopically equivalent to a \((D,V)\)-simplicial complex and whose complement is given by a finite number of connected components which are given by either cusps or disk bundles over \(\mathbb{S}^1\). Here a \((D,V)\)-simplicial complex has at most \(V\) vertices and each vertex has degree at most \(D\). For that kind of simplicial complexes there is a uniform bound given in terms of \((D,V)\) of the logarithm of the cardinality of the torsion part in homology, whence the main statement.
Additionally, by counting the \(2\)-skeleta of such complexes, the authors show that the number \(\Gamma_d(v)\) of (homotopy classes of) \(d\)-dimensional complete negatively curved manifolds with volume less than \(v\) grows asymptotically as \(v\text{log}(v)\).
They conclude by showing that in the \(3\)-dimensional case one cannot control the torsion part studying sequences of real hyperbolic manifolds converging in the Benjamini-Schramm topology.

MSC:

57R19 Algebraic topology on manifolds and differential topology
55N99 Homology and cohomology theories in algebraic topology
55P10 Homotopy equivalences in algebraic topology
22E40 Discrete subgroups of Lie groups
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