On periodic groups of homeomorphisms of the 2-dimensional sphere

TL;DR

This paper proves that finitely-generated groups of sphere homeomorphisms with bounded finite orders are finite, extending to area-preserving cases with even order elements, revealing structural constraints of such groups.

Contribution

It establishes finiteness of certain groups of sphere homeomorphisms under specific order constraints, generalizing previous results.

Findings

Finitely-generated groups with bounded order elements are finite.

Similar finiteness holds for area-preserving homeomorphisms with even order elements.

Provides new insights into the structure of groups acting on the 2-sphere.

Abstract

We prove that every finitely-generated group of homeomorphisms of the 2-dimensional sphere all of whose elements have a finite order which is a power of 2 and so that there exists a uniform bound for the order of group elements is finite. We prove a similar result for groups of area-preserving homeomorphisms without the hypothesis that the orders of group elements are powers of 2 provided there is an element of even order.