Rectangle groups

TL;DR

This paper investigates a specific class of groups with simple presentations, revealing unique properties such as incompatible splittings and actions on geometric structures, challenging existing theories on group decompositions.

Contribution

It introduces a new class of groups with simple presentations that exhibit unexpected properties, providing counterexamples to certain conjectures in group theory.

Findings

Group does not have 2-orbifold or isometry groups as homomorphic images.

Exhibits incompatible splittings over non-small subgroups.

Has an unstable action on an R-tree and a cocompact action on a CAT(0) cube complex.

Abstract

A class of groups is investigated, each of which has a fairly simple presentation . For example the group $R = (a, b, c, d | a^3 = b^3 = c^3 = d^3 = 1, ba^{-1} =dc^{-1}, ca^{-1} = db^{-1}) $ is in the class. Such a group does not have as a homomorphic image any group which is a 2-orbifold group or which is a group of isometries of the reals. However it does have incompatible splittings over subgroups which are not small. This contradicts some ideas I had about universal JSJ decompostions for finitely presented groups over finitely generated subgroups. Such a group also has an unstable action on an R-tree and a cocompact action on a CAT(0) cube complex with finite cyclic point stabilizers, and trivial edge stabilizers.