arXiv research

Study on braid group quotients by congruence subgroups.

Study on braid groups' congruence subgroups and their crystallographic quotients.

The study explores congruence subgroups of braid groups and their quotients.

Proves Congruence Subgroup Property for two types of groups.

Criterion for congruence RFRS towers in hyperbolic lattices.

Proves congruence subgroup property for mapping class groups of hyperbolic surfaces.

New stability results for homology groups of Torelli subgroups and congruence subgroups.

Study virtual braid groups, proving a key subgroup result.

New congruence subgroups found from braid group representations.

The paper studies congruence subgroups and crystallographic quotients of small Coxeter groups.

Enhances stability ranges for Torelli and congruence subgroup homologies.

It is known that the level $2$ principal congruence subgroup of $GL(n;\mathbb{Z})$ has a finite generating set. In this paper, we give a finite presentation of the level $2$ principal congruence subgroup of $GL(n;\mathbb{Z})$.

Improved estimate for remainder term in Weyl law for congruence subgroups of Chevalley groups.

The study characterizes subgroups of braid groups and their finite quotients.

Paper explores relations between braid groups and their quotients.

In this article we present an unpublished proof of W. Thurston that pure braid groups have the congruence subgroup property.

In this paper we prove that for suitable sequences of congruence subgroups of Bianchi groups, including the standard exhaustive sequences of a congruence subgroup, and even symmetric powers of the standard representation of Sl_2(C) the size of the torsion part in the first homology grows exponentially. This extends res…

Unified framework for studying Torelli group and congruence subgroup maps.

By evaluating the Burau representation at t=-1, we obtain a symplectic representation of the braid group. We define the congruence subgroups of the braid group to be the preimages of the principal congruence subgroups of the symplectic group. Our main result is that the level four congruence subgroup of the braid group…

Study of modular representations in homology of congruence subgroups.

Study shows conjugacy of torsion in genus 2 surfaces.

Generators found for a specific subgroup of braid groups.

There is an established bijection between finite-index subgroups Gamma of Gamma(2) and bipartite graphs on surfaces, or, equivalently, certain triples of permutations. We utilize this relationship to study both congruence and noncongruence subgroups in terms of the corresponding graphs. We show some elementary criteria…

Sharp upper bound for FI-module regularity, refining stability of congruence subgroup homology.

We study Veech groups of covering surfaces of primitive translation surfaces. Therefore we define congruence subgroups in Veech groups of primitive translation surfaces using their action on the homology with entries in $\mathbb{Z}/a\mathbb{Z}$. We introduce a congruence level definition and a property of a primitive t…

Let G=SO(n,1) and Gamma a geometrically finite Zariski dense subgroup of G which is contained in an arithmetic subgroup of G. Denoting by Gamma(q) the principal congruence subgroup of Gamma of level q, and fixing a positive number λ_0 strictly smaller than (n-1)^2/4, we show that, as q tends to infinity along primes, t…

Let S be a smooth affine algebraic curve, and let S' be the Riemann surface obtained by removing a point from S. We provide evidence for the congruence subgroup property of the mapping class group Mod(S') by showing that its congruence kernel lies in the centralizer of every braid in Mod(S'). As a corollary, we obtain …

This paper determines a minimal generating set and abelianization of a specific subgroup of SL(n,Z).

Study shows Steinberg representation's multiplicity in cohomology of congruence subgroups.

Study subgroups of Bianchi groups with effective methods.

Totally non congruence Veech groups found in translation surfaces.

We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole, and denoted syspi_1. P. Buser and P. Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in the genus g(X). Th…

Let $Γ<\mathrm{SL}_2(\mathbb{Z})$ be a non-elementary finitely generated subgroup and let $Γ(q)$ be its congruence subgroup of level $q$ for each $q\in \mathbb{N}$. We obtain an asymptotic formula for the matrix coefficients of $L^2(Γ(q) \backslash \mathrm{SL}_2(\mathbb{R}))$ with a {\it uniform} exponential error term…

Let A denote the algebraic closure of the rationals Q in the complex numbers C. Suppose G is a torsion-free group which contains a congruence subgroup as a normal subgroup of finite index and denote by U(G) the C-algebra of closed densely defined unbounded operators affiliated to the group von Neumann algebra. We prove…

Starting from the results in math.DG:1212.3161 we prove that for a given Bianchi group, certain natural coefficent modules and a lot of sequences of congruence subgroups of the size of the torsion subgroup of the first homology grows exponentially with the index (we give an explicit rate). We also prove limit multiplic…

Study top dimensional cohomology groups of congruence subgroups of SL_n(Z).

Computes isotropy subgroups of orthogonal matrices acting on Hermitian matrices.

We give a detailed description of the arithmetic Fuchsian group of the Bolza surface and the associated quaternion order. This description enables us to show that the corresponding principal congruence covers satisfy the bound sys(X) > 4/3 log g(X) on the systole, where g is the genus. We also exhibit the Bolza group a…

We study the congruence problem for subgroups of the modular group that appear as Veech groups of square-tiled surfaces in the minimal stratum of abelian differentials of genus two.

In a previous paper, the author (together with Matthew Emerton) proved that the completed cohomology groups of SL_N(Z) are stable in fixed degree as N goes to infinity (Z may be replaced by the ring O_F of integers of any number field). In this paper, we relate these completed cohomology groups to K-theory and Galois c…

We study "how far away" a finite index subgroup G of SL(2,Z) is from being a congruence group. For this we define its deficiency of being a congruence group. We show that the index of the image of G in SL(2,Z/nZ) is biggest, if n is the general Wohlfahrt level. We furthermore show that the Veech groups of origamis (or …

These are the lecture notes for my course at the 2011 Park City Mathematics Graduate Summer School. The first two lectures covered the basics of the Torelli group and the Johnson homomorphism, and the third and fourth lectures discussed the second cohomology group of the level p congruence subgroup of the mapping class…

The paper proves a free product decomposition for congruence subgroups with constraints.

We calculate the abelianizations of the level $L$ subgroup of the genus $g$ mapping class group and the level $L$ congruence subgroup of the $2g \times 2g$ symplectic group for $L$ odd and $g \geq 3$.

Uniform mixing proved for hyperbolic manifolds using congruence covers.

The abstract discusses various open problems in mapping class groups.

In this paper, we classify all the orientable hyperbolic 5-manifolds that arise as a hyperbolic space form $H^5/Γ$ where $Γ$ is a torsion-free subgroup of minimal index of the congruence two subgroup $Γ^5_2$ of the group $Γ^5$ of positive units of the Lorentzian quadratic form $x_1^2+...+x_5^2-x_6^2$. We also show that…

New construction shows DAAG/DAHA actions on congruence subgroups.