Related papers: Cohomology at Infinity and the Well-Tempered Compl…

We present an algorithm to compute the Hecke operators on the equivariant cohomology of an arithmetic subgroup $\Gamma$ of the general linear group $\mathrm{GL}_n$. This includes $\mathrm{GL}_n$ over a number field or a finite-dimensional…

We study cocycles of countable groups $\Gamma$ of Borel automorphisms of a standard Borel space $(X, \mathcal{B})$ taking values in a locally compact second countable group $G$. We prove that for a hyperfinite group $\Gamma$ the subgroup of…

This note presents a general theorem about the cohomology of finite dimensional Lie algebras of arbitrary characteristic. As an application we compute the cohomology of the Borel subalgebra of sl(N).

Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction (EG x X)/G of a proper G-CW-complex X satisfying certain finiteness conditions. In particular we give formulas…

We extend the computations in [AGM1, AGM2, AGM3] to find the cohomology in degree five of a congruence subgroup Gamma of SL(4,Z) with coefficients in a field K, twisted by a nebentype character eta, along with the action of the Hecke…

Let $N>1$ be an integer, and let $\Gamma = \Gamma_0 (N) \subset \SL_4 (\Z)$ be the subgroup of matrices with bottom row congruent to $(0,0,0,*)\mod N$. We compute $H^5 (\Gamma; \C) $ for a range of $N$, and compute the action of some Hecke…

We prove explicit and elementary formulas for the group homology and cohomology of a finite group with coefficients in any module. We describe in elementary terms the cohomology algebra $H^*(G,k)$ as a graded algebra for a finite group $G$…

This paper describes an approach to computer aided calculations in the cohomology of arithmetic groups. It complements existing literature on the topic by emphasizing homotopies and perturbation techniques, rather than cellular subdivision,…

We prove standard results of group cohomology -- namely, existence of a long exact sequence, classification of torsors via the first cohomology group, Shapiro's lemma, the Hochschild-Serre spectral sequence, a decomposition of the cochain…

In this paper, we generalize Serre's splitting theorem for cohomological invariants of the symmetric group to finite Coxeter groups, provided that the ground field has characteristic zero. We then use this principle to determine all the…

We prove that the completed cohomology groups of SL_N(Z) in fixed degree stabilize as N goes to infinity. We also prove that the action of Hecke operators on stable cohomology is trivial, in a precisely defined sense.

Let $\Gamma$ be a torsion-free arithmetic group acting on its associated global symmetric space $X$. Assume that $X$ is of non-compact type and let $\Gamma$ act on the geodesic boundary $\partial X$ of $X$. Via general constructions in…

We show that the hypercohomology of the Chevalley-Eilenberg-de Rham complex of a Lie algebroid L over a scheme with coefficients in an L-module can be expressed as a derived functor. We use this fact to study a Hochschild-Serre type…

This paper applies the decomposition theorem in intersection cohomology to geometric invariant theory quotients, relating the intersection cohomology of the quotient to that of the semistable points for the action. Suppose a connected…

Let $\Gamma$ be a lattice in a locally compact group $G$. In earlier work, we used $KK$-theory to equip the $K$-groups of any $\Gamma$-$C^{*}$-algebra on which the commensurator of $\Gamma$ acts with Hecke operators. When $\Gamma$ is…

Higher order cohomology of arithmetic groups is expressed in terms of (g,K)-cohomology. Generalizing results of Borel, it is shown that the latter can be computed using functions of (uniform) moderate growth. A higher order versions of…

We prove a version of Quillen's stratification theorem in equivariant homotopy theory for a finite group $G$, generalizing the classical theorem in two directions. Firstly, we work with arbitrary commutative equivariant ring spectra as…

We extend the computations in our prior work to find the cohomology in degree five of a congruence subgroup Gamma of SL_4(Z) with coefficients in Sym^g(K^4), twisted by a nebentype character eta, along with the action of the Hecke algebra.…

Given a finite group action on a (suitably enhanced) triangulated category linear over a field, we establish a formula for the Hochschild cohomology of the category of invariants, assuming the order of the group is coprime to the…

For a smooth quasi-projective surface S over complex numbers we consider the Borel-Moore homology of the stack of coherent sheaves on S with compact support and make this space into an associative algebra by a version of the Hall…