Cohomology at infinity and the well-rounded retract for general Linear Groups

Let $\bold G$ be a reductive algebraic group defined over \Q, and let $\Gamma$ be an arithmetic subgroup of \bold G(\Q). Let $X$ be the symmetric space for $\bold G(\R)$, and assume $X$ is contractible. Then the cohomology (mod torsion) of the space $X/\Gamma$ is the same as the cohomology of $\Gamma$. In turn, $X/\Gamma$ will have the same cohomology as $W/\Gamma$, if $W$ is a ``spine'' in $X$. This means that $W$ (if it exists) is a deformation retract of $X$ by a $\Gamma$-equivariant deformation retraction, that $W/\Gamma$ is compact, and that $\dim W$ equals the virtual cohomological dimension (vcd) of $\Gamma$. Then $W$ can be given the structure of a cell complex on which $\Gamma$ acts cellularly, and the cohomology of $W/\Gamma$ can be found combinatorially

Rigidity of p-adic cohomology classes of congruence subgroups of GL(n, Z)

We extend the work of Ash and Stevens [Ash-Stevens 97] on p-adic analytic families of p-ordinary arithmetic cohomology classes for GL(N,Q) by introducing and investigating the concept of p-adic rigidity of arithmetic Hecke eigenclasses. An arithmetic eigenclass is said to be "rigid" if (modulo twisting) it does not admit a nontrivial p-adic deformation containing a Zariski dense set of arithmetic specializations. This paper develops tools for explicit investigation into the structure of eigenvarieties for GL(N). We use these tools to prove that known examples of non-sefldual cohomological cuspforms for GL(3) are rigid. Moreover, we conjecture that for GL(3), rigidity is equivalent to non-selfduality.Comment: 23 page

Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois representations

We report on the computation of torsion in certain homology theories of congruence subgroups of SL(4,Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2,3,5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels up to 31