Isometry groups of spaces with curvature bounded above (Q1818768)

The following problem is considered: what are the conditions on the group \(G\) of isometries of a metric space \(X\) equipped with the compact open topology to be a Lie group (i.e. the identity component \(G_0\) is a Lie group, and \(G/G_0\) is discrete). The author proves that, for a locally compact length space \(X\) with curvature bounded above and for any compact set \(C \subset int X\), the subgroup of \(\text{Isom} (X)\) leaving \(C\) invariant is a Lie group as a subgroup of \(\text{Isom} (C)\). As a corollary he gets that, if \(X\) is a compact geodesically complete space with curvature bounded above, then \(\text{Isom} (X)\) is a Lie group. Then, the author describes the small subgroups of a compact group of isometries and proves that, if for a locally compact geodesically complete Hadamard space \(X\) the ideal boundary \(X(\infty)\) endowed with the Tits metric \(Td\) is compact, then both, \(\text{Isom} (X)\) and \(\text{Isom} (X(\infty), Td)\), are Lie groups. Finally, the obtained results are applied to spaces of negative curvature. Namely, let \(X\) be a locally compact geodesically complete space with negative curvature satisfying the following: 1) The Hausdorff dimension of \(X\) is finite; 2) The Hausdorff \(n\)-measure of \(X^n\) is finite for every \(n \geq 2\); 3) \(X\) is isometric to neither the real line nor a circle. The author proves that, under these conditions, the group \(\text{Isom} (X)\) is totally disconnected. This is a generalization of the well known result concerning the isometry group of a Riemannian manifold of negative curvature.