Commensurators and Quasi-Normal Subgroups

TL;DR

This paper introduces the concept of quasi-normal subgroups in finitely generated groups, providing geometric characterizations and exploring their properties and differences from normal subgroups.

Contribution

It develops the basic theory of quasi-normal subgroups, including geometric characterizations and their relation to coset graphs, expanding understanding beyond normal subgroups.

Findings

Quasi-normal subgroups are kernels of specific maps.

A subgroup is quasi-normal iff its coset graph is locally finite.

The paper establishes geometric criteria for quasi-normality.

Abstract

We say A is a quasi-normal subgroup of the group G if the commensurator of A in G is all of G. We develop geometric versions of commensurators in finitely generated groups. In particular, g is an element of the commensurator of A in G iff the Hausdorff distance between A and gA is finite. We show that a quasi-normal subgroup of a group is the kernel of a certain map, and a subgroup of a finitely generated group is quasi-normal iff the natural coset graph is locally finite. This last equivalence is particularly useful for deriving asymptotic results for finitely generated groups. Our primary goal in this paper is to develop the basic theory of quasi-normal subgroups, comparing analogous results for normal subgroups and isolating differences between quasi-normal and normal subgroups.