Related papers: Steinberg homology, modular forms, and real quadra…

Let $ G $ be a real simple linear connected Lie group of real rank one. Then, $ X := G/K $ is a Riemannian symmetric space with strictly negative sectional curvature. By the classification of these spaces, $X$ is a real/complex/quaternionic…

We study the asymptotic behaviour of the cohomology of subgroups $\Gamma$ of an algebraic group $G$ with coefficients in the various irreducible rational representations of $G$ and raise a conjecture about it. Namely, we expect that the…

Let ${\cal M}_{g,n}$ and ${\cal H}_{g,n}$, for $2g-2+n>0$, be, respectively, the moduli stack of $n$-pointed, genus $g$ smooth curves and its closed substack consisting of hyperelliptic curves. Their topological fundamental groups can be…

If $R$ is a commutative ring, $M$ a compact $R$-oriented manifold and $G$ a finite graph without loops or multiple edges, we consider the graph configuration space $M^G$ and a Bendersky-Gitler type spectral sequence converging to the…

If $\Gamma$ is a finitely generated Fuchsian group such that its derived subgroup $\Gamma'$ is co-compact and torsion free, then $S={\mathbb H}^{2}/\Gamma'$ is a closed Riemann surface of genus $g \geq 2$ admitting the abelian group…

For $R$ a Euclidean number ring, and let $\Gamma_n(p)$ be the level-$p$ principal congruence subgroup of $\text{SL}_n(R)$. Borel--Serre showed that the cohomology of $\Gamma_n(p)$ vanishes above a degree $\nu$ that is quadratic in $n$. Let…

The groups $\Gamma_{n,s}$ are defined in terms of homotopy equivalences of certain graphs, and are natural generalisations of $\mbox{Out}(F_n)$ and $\mbox{Aut}(F_n)$. They have appeared frequently in the study of free group automorphisms,…

We prove new homological stability results for general linear groups over finite fields. These results are obtained by constructing CW approximations to the classifying spaces of these groups, in the category of $E_\infty$-algebras, guided…

Let E/Q be a real quadratic field and pi_0 a cuspidal, irreducible, automorphic representation of GL(2,A_E) with trivial central character and infinity type (2,2n+2) for some non-negative integer n. We show that there exists a non-zero…

The goal of this paper is to construct and describe certain arithmetic subgroups of the automorphism group of a partially commutative group. More precisely, given an arbitrary finite graph $\Gamma$ we construct an arithmetic subgroup…

The symmetric homology of a unital algebra $A$ over a commutative ground ring $k$ is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring $A = k[\Gamma]$, the symmetric homology is related to…

We show that noncongruence subgroups of SL_2(Z) projectively equivalent to congruence subgroups are ubiquitous. More precisely, they always exist if the congruence subgroup in question is a principal congruence subgroup Gamma(N) of level…

We define the space of nearly holomorphic automorphic forms on a connected reductive group $G$ over $\mathbb{Q}$ such that the homogeneous space $G(\mathbb{R})^1/ K_\infty^\circ$ is a Hermitian symmetric space. By Pitale, Saha and Schmidt's…

We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair (G,K), where G=Sp_4(R) and K is its maximal compact subgroup. In particular, we determine K-types and composition series, and write down explicit…

We give several resolutions of the Steinberg representation St_n for the general linear group over a principal ideal domain, in particular over Z. We compare them, and use these results to prove that the computations in [AGM4] are…

Let $\Gamma$ be a word hyperbolic group with a cyclic JSJ decomposition that has only rigid vertex groups, which are all fundamental groups of closed surface groups. We show that any group $H$ quasi-isometric to $\Gamma$ is abstractly…

Let $\ell \geq 5$ be a prime, and let $\nu_\eta$ denote the Dedekind eta multiplier. For an odd integer $r$, and a real Dirichlet character $\psi$, recent work of Ahlgren, Andersen, and the author showed that quadratic congruences modulo…

Suppose that $\ell \geq 5$ is prime. For a positive integer $N$ with $4 \mid N$, previous works studied properties of half-integral weight modular forms on $\Gamma_0(N)$ which are supported on finitely many square classes modulo $\ell$, in…

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 study homological invariants of the Steinberg algebra $\mathcal{A}_k(\mathcal{G})$ of an ample groupoid $\mathcal{G}$ over a commutative ring $k$. For $\mathcal{G}$ principal or Hausdorff with…