Commensurators of non-free finitely generated Kleinian groups

TL;DR

This paper investigates the structure of commensurators of certain non-free finitely generated Kleinian groups, establishing conditions under which these commensurators are discrete or lattices, depending on the group's properties.

Contribution

It proves that for non-free finitely generated Kleinian groups without parabolics and not lattices, the commensurator is discrete unless the group is a fiber group, where it becomes a lattice.

Findings

C(G) is discrete if the limit set is not a round circle.

G has finite index in C(G) unless G is a fiber group.

C(G) is a lattice when G is a fiber group.

Abstract

Suppose G is a non-free finitely generated Kleinian group without parabolics which is not a lattice and let C(G) denote the commensurator in PSL(2,C). We prove that if the limit set of G is not a round circle, then C(G) is discrete. Furthermore, G has finite index in C(G) unless G is a fiber group in which case C(G) is a lattice.