I will describe joint work with N. Nikolov showing
that the universal lattices have property tau.
These are the first examples of non arithmetic
groups with this property. In particular we show that
the groups $SL_d(Z[x_1,…,x_k]S have property tau.
Almost the whole proof can be generalized to the
non-commutative analogs of these groups.
This generalization leads to several interesting applications:
The Cayley graphs almost all finite simple groups
can be made expanders using suitable generating sets.
This allows us tho disprove an old conjecture
of Lubotzky and Wiess that an amenable group and
a group with property tau can not be simultaneously
dense subgroups of an infinite compact group.
This construction also gives us a finite
generated group having tau with very large
subgroup growth.