On the number of finite subgroups of a lattice

TL;DR

This paper establishes bounds on the number of finite subgroups in lattices of semisimple Lie groups, showing linear bounds in general and sublinear growth in higher rank cases, with implications for orbifold stratifications.

Contribution

It provides new bounds on the number of finite subgroups in lattices, revealing linear and sublinear growth patterns depending on the group rank.

Findings

Number of conjugacy classes of maximal finite subgroups is linearly bounded by covolume.

In higher rank groups, this number grows sublinearly with covolume.

Results imply volume bounds for strata in locally symmetric orbifolds.

Abstract

We show that the number of conjugacy classes of maximal finite subgroups of a lattice in a semisimple Lie group is linearly bounded by the covolume of the lattice. Moreover, for higher rank groups, we show that this number grows sublinearly with covolume. We obtain similar results for isotropy subgroups in lattices. Geometrically, this yields volume bounds for the number of strata in the natural stratification of a finite-volume locally symmetric orbifold.