Machine generated contents note: Introduction; 1. First and second variational formulas for
area; 2. Volume comparison theorem; 3. Bochner-Weitzenböck formulas; 4. Laplacian …

We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral
invariants of sequences of locally symmetric spaces. Our main results are uniform versions …

The systole of a compact metric space $ X $ is a metric invariant of $ X $, defined as the
least length of a noncontractible loop in $ X $. When $ X $ is a graph, the invariant is usually …

Let $ G $ be a higher rank semisimple real Lie group or the product of at least two
automorphism groups of regular trees. We prove all probability measure preserving actions …

In the mid 60's, D. Kazhdan defined his Property (T) for locally compact groups and used it
as a tool to demonstrate that a large class of lattices are finitely generated. Recall that a …

A classical theorem of Wang [W] implies that for a fixed dimension n≥ 4, and any V∈ R,
there are only finitely many complete hyperbolic manifolds without boundary of volume at …

We show that for every n≥ 2 and any ϵ> 0 there exists a compact hyperbolic n–manifold
with a closed geodesic of length less than ϵ. When ϵ is sufficiently small these manifolds …

We show that for any non-virtually solvable finitely generated group of matrices over any
field, there is an integer m such that, given an arbitrary finite generating set for the group …

We announce new results concerning the asymptotic behavior of the Betti numbers of higher
rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a …

We find an upper bound for the entropy of a systolically extremal surface, in terms of its
systole. We combine the upper bound with Katok's lower bound in terms of the volume, to …