A very wide array of properties are compatible with these hypotheses. Burger--Mozes famously gave examples of infinite simple groups of this form. Earlier, Wise and Bhattacharjee had independently given examples of such groups with no proper finite quotients. In particular, these groups have no non-trivial virtually nilpotent quotients.
On the other hand, one can construct examples of such groups with many proper quotients (using various flavours of the Rips construction, for instance -- see Bridson and Haefliger for details).
So I think the answer to your question is that not much can be said without further information.