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Consider a product of two locally-finite, infinite, unimodular trees $X=T_1\times T_2$. Assume that both ${\rm Aut}(T_1)$ and ${\rm Aut}(T_2)$ are not discrete.

So as a vague general question, what is known about non-uniform lattices in ${\rm Isom}(X)$?

Clearly, if one of the $T_i$s admit a non-uniform lattice $\Gamma_1$ (say $T_1$) then we can take any lattice $\Gamma_2$ in $T_2$ and the direct product $\Gamma=\Gamma_1\times\Gamma_2$ will be a non-uniform lattice in ${\rm Isom}(X)$. However, this lattice is reducible (both as a lattice or an abstract group).

A more specific question is as follows. What are examples of non-uniform lattices with non-discrete projection to $T_1$ and $T_2$? What is known about them (e.g. density of the commensurators, residual finiteness etc.)?

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  • $\begingroup$ "Product of trees": the isometry group might depend on the precise meaning of "product". Also, for a tree, unimodular means the automorphism group is unimodular? $\endgroup$
    – YCor
    Nov 10 '19 at 17:25
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    $\begingroup$ @YCor 'product' meaning the sensible one that gives you a ${\rm CAT}(0)$-space. And yes unimodular tree meaning the automorphism group is unimodular. $\endgroup$
    – Sam Hughes
    Nov 10 '19 at 19:59
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    $\begingroup$ I'm going to once again suggest that you might like to ask Pierre-Emmanuel Caprace! He wrote a very nice set of notes on lattices in products of tree, based on a mini-course at the Newton Institute: arxiv.org/abs/1709.05949 . However, a cursory search suggests that his notes don't say much about the non-uniform case. $\endgroup$
    – HJRW
    Nov 14 '19 at 11:16
  • $\begingroup$ @HJRW Thank you for the notes, I think I may have to email him! $\endgroup$
    – Sam Hughes
    Nov 14 '19 at 15:17

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