Commensurators of some non-uniform tree lattices and Moufang twin trees

TL;DR

This paper investigates the density of commensurators in automorphism groups of certain non-uniform tree lattices, extending known results to a broader class including those from Moufang twin trees, with implications for arithmeticity.

Contribution

It generalizes the density of commensurators to many Nagao type lattices and Moufang twin trees, broadening the scope of previous results.

Findings

Density of commensurators holds for many Nagao type lattices

Results include lattices derived from Moufang twin trees

Supports generalized arithmeticity in these contexts

Abstract

Sh. Mozes showed that the commensurator of the lattice ${\rm PSL}_2 \bigl({\bf F}_p[t{}^{-1}] \bigr)$ is dense in the full automorphism group of the Bruhat-Tits tree of valency $p+1$, the latter group being much bigger than ${\rm PSL}_2 \bigl({\bf F}_p((t)) \bigr)$. By G.A. Margulis' criterion, this density is a generalized arithmeticity result. We show that the density of the commensurator holds for many tree-lattices among those called of Nagao type by H. Bass and A. Lubotzky. The result covers many lattices obtained via Moufang twin trees.