In joint work with Yehuda Shalom, we have proved Margulis'
Normal Subgroup Theorem for any discrete subgroup $\Gamma$ of the
automorphism group of a locally finite $A_2$-tilde building, $B$,
provided that the quotient of $B$ by $\Gamma$ is compact...
We consider maps between smooth projective curves and some
arithmetic and geometric properties of such maps. In particular, we
will discuss the case of maps from the generic Riemann surface of
genus g -- a problem first seriously looked at by...
Let $G$ be a simple $G(\mathbf Q)$-group of $G(\mathbf Q)$-rank
at least 2. In 1987 T. N. Venkataramana showed that if $\Gamma
\subset G(\mathbf Z)$ is an infinite subgroup whose commensurator
is a subgroup of finite index in $G(\mathbf Z)$, then $...
A pair (G,H), where G is a group and H a subgroup, has relative
Property T if every isometric action of G on a Hilbert space has a
H-fixed point. In a connected Lie group or a lattice G, we
characterize subgroups H such that (G,H) has relative...
The remarkable phenomenon of Superrigidity, discovered by
Margulis in the context of linear representations of lattices in
higher rank semi-simple groups, has motivated and inspired a lot of
research on other "higher rank" groups and representations...
We analyze volume-preserving actions of product groups on
Riemannian manifolds. Under a natural spectral irreducibility
assumption, we prove the following dichotomy: Either the action is
measurably isometric, in which case there are at most two...
In previous work we showed that arithmetic hyperbolic
2-manifolds that are isospectral are commensurable. In this talk we
discuss the proof of the generalization to dimension 3. We had
previously shown that if arithmetic hyperbolic 3-manifolds
are...
In this talk I will give an overview of joint work with M.
Rapoport and T. Yang on the construction of generating series whose
coefficients are the classes of special divisors and 0-cycles on
the arithmetic surfaces attached to Shimura curves. These...
We obtain asymptotic lower bounds for the spectral function of
the Laplacian and for the remainder in local Weyl's law on compact
manifolds. In the negatively curved case, thermodynamic formalism
is applied to improve the estimates. Our results can...
