Arithmetic Kleinian groups generated by elements of finite order

TL;DR

This paper proves that, up to commensurability, only finitely many cocompact arithmetic Kleinian groups generated by finite order rotations exist, using geometric inequalities and volume bounds.

Contribution

It establishes finiteness results for cocompact arithmetic Kleinian groups generated by elements of finite order, extending understanding of their classification.

Findings

Finiteness of cocompact arithmetic Kleinian groups generated by rotations

Finiteness of conjugacy classes of two-generated arithmetic Kleinian groups

Application of generalized Gromov--Guth inequality and volume bounds

Abstract

We show that up to commensurability there are only finitely many cocompact arithmetic Kleinian groups generated by rotations. This implies, in particular, that there exist only finitely many conjugacy classes of cocompact two generated arithmetic Kleinian groups. The proof of the main result is based on a generalized Gromov--Guth inequality and bounds for the hyperbolic and tube volumes of the quotient orbifolds.