Sieve methods in group theory I: Powers in Linear groups

TL;DR

This paper introduces a general sieve method for finitely generated groups and demonstrates that in certain linear groups, the set of proper powers is exponentially rare, significantly strengthening previous results.

Contribution

It formulates a new sieve technique for groups and applies it to show proper powers are exponentially rare in non virtually-solvable linear groups.

Findings

Proper powers form an exponentially small subset in the specified groups.

The sieve method effectively measures subset sizes within finitely generated groups.

Strengthens previous results on the distribution of powers in linear groups.

Abstract

A general sieve method for groups is formulated. It enables one to "measure" subsets of a finitely generated group. As an application we show that if $\Gamma$ is a finitely generated non virtually-solvable linear group of characteristic zero then the set of proper powers in $\Gamma$ is exponentially small. This is a far reaching strengthening of the main result of \cite{HKLS}.