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

### Angela Kubena and Anne Thomas

#### Abstract

Let $G$ be the automorphism group of a regular right-angled building $X$. The "standard uniform lattice" $\Gamma_0$ in $G$ 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 $\Gamma_n$ in $G$, 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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 Wednesday, September 22, 2010