Lattice actions, 3-manifolds and homology
An action \(\psi: \Gamma\rightarrow \text{Homeo}(M)\) of a group \(\Gamma\) on a compact \(n\)-manifold \(M\) induces an action on the rational homology of \(M\): \(\psi^i_*: \Gamma\rightarrow \text{Aut}(H_i(M;\mathbb{Q}))\) for each \(0\leq i\leq n\). If \(\psi^i_*(\Gamma)\) is finite for each \(i\), the action is called homologically finite. Otherwise is is called homologically infinite. In Theorem 1, the following statements are proved: Let \(\Gamma\) be an irreducible lattice in a semisimple Lie group of \(\mathbb{R}\)-rank at least 2, and let \(M\) be a closed, orientable, irreducible, connected \(3\)-manifold. If \(\psi\) is homologically infinite action, then: (1) \(M\) is homeomorphic to the \(3\)-torus \(T^3\). (2) \(\Gamma \) is a finite-index subgroup of (a conjugate of) \(SL(3,\mathbb{Z})\). (3) \(\psi_*\) is conjugate to the restriction to \(\Gamma\) of the standard action of \(SL(3,\mathbb{Z})\) on \(H_1(T^3;\mathbb{Q})\). On fact: for each element \(\gamma \in \Gamma\), the homeomorphism \(\psi(\gamma)\) is isotopic to the standard linear action of \(\gamma \) on \(T^3\). Homologically infinite lattice actions on other (necessarily reducible) \(3\)-manifolds do exist, and Theorems 2 and 3 of this paper give strong restrictions on such actions.
scientific article; zbMATH DE number 1436053