In a paper from 1973, Garland gave criteria to the vanishing of $L^2$ cohomologies of lattices p-adic groups by studying their action on Tits buildings. Later, Ballmann and Swiatkowski and independently Zuk generalized those criteria to groups acting on a simplicial complex. Those criteria regarded only the geometrical properties of the simplicial complex as long as the group action was well enough (for instance if the group acted properly discontinuously and cocompactly).
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The most famous of those results, was Zuk’s criterion for the vanishing of the first $L^2$ cohomology (which is equivalent to property (T) when the group is locally compact with a countable base). Namely, Zuk proved that a group acting well on a two dimensional simplicial complex will have property (T) if the first positive eigenvalue of the graph Laplacian at the link of each vertex of the complex will be strictly larger than 1/2.
In my talk, I’ll present some geometrical intuition to Zuk’s criterion, present two new criteria for (T) property (one of them being a generalization of Zuk’s criterion) and give some examples. If time permits, I’ll mention how those results pass to a group acting on simplicial complex of dimension larger than 2 and present a new vanishing result for all the cohomologies.