Murray State Theses and Dissertations

Abstract

Associated to any graph Γ are several groups where their presentations are encoded by Γ. Two such groups are right-angled Coxeter groups and right-angled Artin groups. By introducing more structure to Γ, what we call "local involutions," the graph Γ becomes a right-angled mock reflection system and encodes the presentation of two more groups: right-angled mock reflection groups and right-angled mock Artin groups.

Here we show that every right-angled mock Artin group associated with an n-gon graph with local involutions is a finite index subgroup in some right-angled mock reflection group. We employ a strategy similar to the one Davis and Januszkiewicz employ to show all right-angled Artin groups are finite index in some right-angled Coxeter group. In ours, an n-gon graph with local involutions Γ is mapped onto a finite group with graph Γ0, and that map is used to create a graph with local involutions Γ` by pairing the vertices of Γ with those of Γ0 they are not mapped to. The right-angled mock reflection group associated with Γ` is the one in which the right-angled mock Artin group associated with Γ is finite index. This is proven by finding an isomorphism β from the right-angled mock Artin group to a subgroup of W` that is known to be of finite index.

Year manuscript completed

2022

Year degree awarded

2022

Author's Keywords

mock reflection groups, mock Artin groups

Thesis Advisor

Timothy Schroeder

Committee Member

Dubravko Ivansic

Committee Member

Robert Donnelly

Document Type

Thesis

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Algebra Commons

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