Projektdetails
Abstract
Wider research context
We study infinite discrete groups as geometric objects, and draw algebraic conclusions from their geometric structure. Key questions include whether two groups are geometrically equivalent (quasiisometric) to one another, or to some group from a particular family. We construct well-behaved group actions on hyperbolic or non-positively curved metric spaces.
Objectives
Determine when a right-angled Coxeter group is quasiisometric to a right-angled Artin group. Determine when two Coxeter groups are quasiisometric.
Develop computer tools to decide these questions.
Show that every Coxeter group admits a universal acylindrical action on some hyperbolic space.
Show that Coxeter groups are hierarchically hyperbolic.
Approach
For the quasiisometry problem we develop new quasiisometry invariants based on canonical actions of groups coming from their coarse geometry. One such action comes from JSJ decompositions, and we will develop more from higher dimensional splittings. Another such action comes from making a graph from the product region structure, which has been used successfully in right-angled Artin groups and is analogous to the curve complex machinery from mapping class groups. We will extend this analogy to Coxeter groups, including in the non-right-angled case.
Level of originality
Our ideas for constructing quasiisometry invariants are finer than those previously studied. Others have considered that the quasiisometry types of pieces of the JSJ decomposition give invariants, but the further insight that the quasiisometry types should be considered relative to peripheral patterns has not been exploited. Others have observed that the product region graph of a right-angled Artin group is a quasitree, but we find even finer structure that distinguishes between the type of quasitree that actually occurs for RAAGs and those that can occur for RACGs.
Previous work generalizing the curve complex machinery from mapping class groups has focused on cubical groups, where wall structures are in certain senses well-behaved, for instance, satisfying the Helly Property.
This fails for the wall structures of Coxeter groups, and prevents one from making an argument for Coxeter groups by first making them act on a cube complex and then applying the construction to the cube complex.
Our innovation in this case is to work directly with the wall structure of the Coxeter group, and show that, for the specific constructions we are interested, the complications are surmountable.
Primary researchers involved
Project Leader: Christopher H. Cashen, PhD Privatdozent
We study infinite discrete groups as geometric objects, and draw algebraic conclusions from their geometric structure. Key questions include whether two groups are geometrically equivalent (quasiisometric) to one another, or to some group from a particular family. We construct well-behaved group actions on hyperbolic or non-positively curved metric spaces.
Objectives
Determine when a right-angled Coxeter group is quasiisometric to a right-angled Artin group. Determine when two Coxeter groups are quasiisometric.
Develop computer tools to decide these questions.
Show that every Coxeter group admits a universal acylindrical action on some hyperbolic space.
Show that Coxeter groups are hierarchically hyperbolic.
Approach
For the quasiisometry problem we develop new quasiisometry invariants based on canonical actions of groups coming from their coarse geometry. One such action comes from JSJ decompositions, and we will develop more from higher dimensional splittings. Another such action comes from making a graph from the product region structure, which has been used successfully in right-angled Artin groups and is analogous to the curve complex machinery from mapping class groups. We will extend this analogy to Coxeter groups, including in the non-right-angled case.
Level of originality
Our ideas for constructing quasiisometry invariants are finer than those previously studied. Others have considered that the quasiisometry types of pieces of the JSJ decomposition give invariants, but the further insight that the quasiisometry types should be considered relative to peripheral patterns has not been exploited. Others have observed that the product region graph of a right-angled Artin group is a quasitree, but we find even finer structure that distinguishes between the type of quasitree that actually occurs for RAAGs and those that can occur for RACGs.
Previous work generalizing the curve complex machinery from mapping class groups has focused on cubical groups, where wall structures are in certain senses well-behaved, for instance, satisfying the Helly Property.
This fails for the wall structures of Coxeter groups, and prevents one from making an argument for Coxeter groups by first making them act on a cube complex and then applying the construction to the cube complex.
Our innovation in this case is to work directly with the wall structure of the Coxeter group, and show that, for the specific constructions we are interested, the complications are surmountable.
Primary researchers involved
Project Leader: Christopher H. Cashen, PhD Privatdozent
| Kurztitel | Coxeter groups |
|---|---|
| Status | Laufend |
| Tatsächlicher Beginn/ -es Ende | 23/11/24 → 22/11/28 |