Is it true that the fundamental groups of compact Kahler surfaces have property T if and only if it they are finite? I am having trouble finding counterexamples to this, but maybe that's just me...

  • $\begingroup$ Igor, if you add the assumption that the fundamental group is Gromov-hyperbolic, then it becomes a very interesting question to which currently there are no counter-examples. $\endgroup$ – Misha Mar 25 '12 at 13:55
  • $\begingroup$ @Misha: Isn't it hard just to find a hyperbolic group with prop T? $\endgroup$ – Igor Rivin Mar 25 '12 at 17:54
  • $\begingroup$ @Igor Rivin: Igor, there are several ways to construct hyperbolic groups with property T. The oldest: (1) Uniform lattices in quaternionic hyperbolic space. More recent: (2) Fundamental groups of 2-dimensional simplicial complexes where links of vertices have smallest eigenvalue $>1/2$. (3) Uniform lattices acting on some hyperbolic buildings. (4) Random groups (in certain regimes) are infinite hyperbolic with property T. Very recent: (5) Oppenheim's constructions. However, it is conjectured that 2-dimensional (hyperbolic) groups are never Kahler (except for surface groups). $\endgroup$ – Misha Mar 25 '12 at 19:54

According to this survey by Donu Arapura, Toledo proved that many arithmetic lattices in higher rank algebraic $\mathbb{Q}$-groups (with hermitian symmetric space) are fundamental groups of smooth projective surfaces.

In particular $Sp(2n,\mathbb{Z})$ for $n>2$, is such a group, and has property (T).

Note that once you get a group as fundamental group of a smooth projective variety you obtain a smooth projective surface with the same fundamental group by intersecting with some generic hyperplanes.

  • $\begingroup$ Ah, thanks! That certainly does it... $\endgroup$ – Igor Rivin Mar 25 '12 at 17:54

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