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Register a new userPlanarity of Cayley graphs of graph products of groups
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Authors
Olga Varghese
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We obtain a complete classification of graph products of finite abelian groups whose Cayley graphs with respect to the standard presentations are planar.
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We show that if $G$ is a group and $G$ has a graph-product decomposition with finitely-generated abelian vertex groups, then $G$ has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly-indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely-generated abelian group and the graph satisfies the $T_0$ property. Our results build on results by Droms, Laurence and Radcliffe.
Given a finite simplicial graph $Gamma=(V,E)$ with a vertex-labelling $varphi:Vrightarrowleft{text{non-trivial finitely generated groups}right}$, the graph product $G_Gamma$ is the free product of the vertex groups $varphi(v)$ with added relations that imply elements of adjacent vertex groups commute. For a quasi-isometric invariant $mathcal{P}$, we are interested in understanding under which combinatorial conditions on the graph $Gamma$ the graph product $G_Gamma$ has property $mathcal{P}$. In this article our emphasis is on number of ends of a graph product $G_Gamma$. In particular, we obtain a complete characterization of number of ends of a graph product of finitely generated groups.
We study coherence of graph products and Coxeter groups and obtain many results in this direction.
Let $G$ be a group and $Ssubseteq G$ its subset such that $S=S^{-1}$, where $S^{-1}={s^{-1}mid sin S}$. Then {it the Cayley graph ${rm Cay}(G,S)$} is an undirected graph $Gamma$ with the vertex set $V(Gamma)=G$ and the edge set $E(Gamma)={(g,gs)mid gin G, sin S}$. A graph $Gamma$ is said to be {it integral} if every eigenvalue of the adjacency matrix of $Gamma$ is integer. In the paper, we prove the following theorem: {it if a subset $S=S^{-1}$ of $G$ is normal and $sin SRightarrow s^kin S$ for every $kin mathbb{Z}$ such that $(k,|s|)=1$, then ${rm Cay}(G,S)$ is integral.} In particular, {it if $Ssubseteq G$ is a normal set of involutions, then ${rm Cay}(G,S)$ is integral.} We also use the theorem to prove that {it if $G=A_n$ and $S={(12i)^{pm1}mid i=3,dots,n}$, then ${rm Cay}(G,S)$ is integral.} Thus, we give positive solutions for both problems 19.50(a) and 19.50(b) in Kourovka Notebook.
A connected, locally finite graph $Gamma$ is a Cayley--Abels graph for a totally disconnected, locally compact group $G$ if $G$ acts vertex-transitively with compact, open vertex stabilizers on $Gamma$. Define the minimal degree of $G$ as the minimal degree of a Cayley--Abels graph of $G$. We relate the minimal degree in various ways to the modular function, the scale function and the structure of compact open subgroups. As an application, we prove that if $T_d$ denotes the $d$-regular tree, then the minimal degree of ${rm Aut}(T_d)$ is $d$ for all $dgeq 2$.
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