Examples of non-uniform lattices in products of trees

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.)?

• "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?
– YCor
Nov 10 '19 at 17:25
• @YCor 'product' meaning the sensible one that gives you a ${\rm CAT}(0)$-space. And yes unimodular tree meaning the automorphism group is unimodular. Nov 10 '19 at 19:59
• 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.
– HJRW
Nov 14 '19 at 11:16
• @HJRW Thank you for the notes, I think I may have to email him! Nov 14 '19 at 15:17