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Register a new userProblems on automorphism groups of nonpositively curved polyhedral complexes and their lattices
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The goal of this paper is to present a number of problems about automorphism groups of nonpositively curved polyhedral complexes and their lattices, meant to highlight possible directions for future research.
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We show that word hyperbolicity of automorphism groups of graph products $G_Gamma$ and of Coxeter groups $W_Gamma$ depends strongly on the shape of the defining graph $Gamma$. We also characterized those $Aut(G_Gamma)$ and $Aut(W_Gamma)$ in terms of $Gamma$ that are virtually free.
The superextension $lambda(X)$ of a set $X$ consists of all maximal linked families on $X$. Any associative binary operation $*: Xtimes X to X$ can be extended to an associative binary operation $*: lambda(X)timeslambda(X)tolambda(X)$. In the paper we study isomorphisms of superextensions of groups and prove that two groups are isomorphic if and only if their superextensions are isomorphic. Also we describe the automorphism groups of superextensions of all groups of cardinality $leq 5$.
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We show that the automorphism group of a graph product of finite groups $Aut(G_Gamma)$ has Kazhdans property (T) if and only if $Gamma$ is a complete graph.
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