
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 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.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Geometric and Algebraic Topology · Finite Group Theory Research
