Golod-Shafarevich groups with property $(T)$ and Kac-Moody groups

Mikhail Ershov
University of Virginia

Informally speaking, a finitely generated group is called Golod-Shafarevich if it has a presentation with a ``small'' set of relators. In 1964, Golod and Shafarevich proved that groups satisfying such condition are necessarily infinite and used this criterion to solve two outstanding problems: the construction of infinite finitely generated periodic groups and the construction of infinite Hilbert class field towers. An important class of Golod-Shafarevich groups consists of the fundamental groups of compact hyperbolic 3-manifolds or, equivalently, torsion-free lattices in $SO(3,1)$. In 1983, Lubotzky used this
fact to prove that arithmetic lattices in $SO(3,1)$ do not have the congruence subgroup property. More recently, Lubotzky and Zelmanov proposed a group-theoretic approach (based on
Golod-Shafarevich techniques) to an even more ambitious problem, Thurston's virtual positive Betti number conjecture. This approach led to the
following question: is it true that Golod-Shafarevich groups never

have property $(\tau)$? I will show that the answer to the above question is

negative in general and describe examples of

Golod-Shafarevich groups with property $(\tau)$ (in fact, property $(T)$)

which are given by lattices in certain topological Kac-Moody groups over finite fields.

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