Density of commensurators for uniform lattices of right-angled buildings

Details Density of commensurators for uniform lattices of right-angled buildings Density of commensurators for uniform lattices of right-angled buildings

2008-12-12

arxiv.org/abs/0812.2280v1

Mathematics

Group Theory

29 pages, 10 figures

Scientific paper

Let $G$ be the automorphism group of a regular right-angled building $X$. The "standard uniform lattice" $\Gamma_0 \leq G$ is a canonical graph product of finite groups, which acts discretely on $X$ with quotient a chamber. We prove that the commensurator of $\Gamma_0$ is dense in $G$. For this, we develop a technique of "unfoldings" of complexes of groups. We use unfoldings to construct a sequence of uniform lattices $\Gamma_n \leq G$, each commensurable to $\Gamma_0$, and then apply the theory of group actions on complexes of groups to the sequence $\Gamma_n$. As further applications of unfoldings, we determine exactly when the group $G$ is nondiscrete, and we prove that $G$ acts strongly transitively on $X$.

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