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.
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Year degree awarded
mock reflection groups, mock Artin groups
Marcum, Zachary, "Finite Index Right-Angled Mock Artin Groups in Right-Angled Mock Reflection Groups" (2022). Murray State Theses and Dissertations. 246.