Home / Tits Alternatives for Graph Products

Yago Antolín (2014)

# Tits Alternatives for Graph Products

In: Extended Abstracts Fall 2012, ed. by González-Meneses, Juan and Lustig, Martin and Ventura, Enric, pp. 1–5, Springer International Publishing. Trends in Mathematics. (ISBN: 978-3-319-05487-2, 978-3-319-05488-9).

Graph ProductsLet Γ=(V,E)Gamma = (V,E) be a simplicial graph and suppose that G=\Gv∣v∈V\mathfrak\G\ =\ G\_\v\mid v in V \ is a collection of groups (called vertex groups). The graph product ΓGGamma mathfrak\G\ of this collection of groups with respect to ΓGamma is the group obtained from the free product of the Gv for v ∈ V by adding the relations [gv,gu]=1 for all gv∈Gv, gu∈Gu such that \v,u\ is an edge of Γ.displaystyle\mbox\ \\$[g\_\v\,g\_\u\] = 1\\$ for all \\$g\_\v\ in G\_\v\\\$, \\$g\_\u\ in G\_\u\\\$ such that \\$\v,u\\\$ is an edge of \\$Gamma \\$\.\ The graph product of groups is a natural group-theoretic construction generalizing free products (when ΓGamma has no edges) and direct products (when ΓGamma is a complete graph) of groups Gv with v ∈ V. Graph products were first introduced and studied by E. Green in her Ph.D. thesis .Basic examples of graph products are right angled Artin groups, also called graph groups (when all vertex groups are infinite cyclic), and right angled Coxeter groups (when all vertex groups are cyc

Group Theory and Generalizations

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