Selberg’s lemma is a fundamental result about linear groups. It states that every finitely generated subgroup of \(\mathrm{GL}(n,K)\), where \(K\) is a field of characteristic zero, is virtually torsion-free (i.e., contains a torsion-free subgroup of finite index).
Recently, Michael Kapovich proved that the conclusion of Selberg’s lemma can fail for finitely generated, discrete subgroups of isometry groups of Hadamard manifolds (arXiv:1808.01602). Recall that a Hadamard manifold is a simply-connected, complete Riemannian manifold of non-positive sectional curvature.
Now what’s the relation of Kapovich’s result and Selberg’s lemma? Let us answer this in the case that the field \(K\) are the real numbers. Consider the maximal compact subgroup \(\mathrm{O}(n,\mathbb{R})\) of \(\mathrm{GL}(n,\mathbb{R})\) and form the homogeneous space \(\mathrm{GL}(n,\mathbb{R}) / \mathrm{O}(n,\mathbb{R})\). It comes with a natural Riemannian metric which turns it into a Hadamard manifold. Hence, if we have a finitely generated subgroup of \(\mathrm{GL}(n,\mathbb{R})\), then it acts on this Hadamard manifold by isometries. And now one can ask if it is actually important for Selberg’s lemma that the Hadamard manifold is exactly the homogeneous space \(\mathrm{GL}(n,\mathbb{R}) / \mathrm{O}(n,\mathbb{R})\). As Michael Kapovich shows, it is important.