Related papers: Explicit growth and expansion for SL_2

A theorem of Elekes and Szab\'{o} recognizes algebraic groups among certain complex algebraic varieties with maximal size intersections with finite grids. We establish a generalization to relations of any arity and dimension, definable in:…

We prove an algebra property under pointwise multiplication for Besov spaces defined on Lie groups of polynomial growth. When the setting is restricted to the case of H-type groups, this algebra property is generalized to paraproduct…

Let $K= \mathbb{Q}(\sqrt{d})$ be a real quadratic field with $d$ having three distinct prime factors. We show that the $2$-class group of each layer in the $\mathbb{Z}_2$-extension of $K$ is $\mathbb{Z}/2\mathbb{Z}$ under certain elementary…

This is a survey of several exciting recent results in which techniques originating in the area known as additive combinatorics have been applied to give results in other areas, such as group theory, number theory and theoretical computer…

We present simple graph-theoretic characterizations of Cayley graphs for monoids, semigroups and groups. We extend these characterizations to commutative monoids, semilattices, and abelian groups.

The l-adic parabolic cohomology groups attached to noncongruence subgroups of SL_2(Z) are finite-dimensional representations of Gal(Qbar/F) for some number field F. We exhibit examples (with F=Q) giving rise to Galois representations whose…

We present a construction of expander graphs obtained from Cayley graphs of narrow ray class groups, whose eigenvalue bounds follow from the Generalized Riemann Hypothesis. Our result implies that the Cayley graph of (Z/qZ)* with respect to…

This paper gives asymptotic formulas for the subgroup growth and maximal subgroup growth of all Baumslag-Solitar groups.

Suppose $X$ and $Y$ are finite complexes, with $Y$ simply connected. Gromov conjectured that the number of mapping classes in $[X,Y]$ which can be realized by $L$-Lipschitz maps grows asymptotically as $L^\alpha$, where $\alpha$ is an…

Let $F$ be a non-Archimedean locally compact field. Let $\sigma$ and $\tau$ be finite-dimensional semisimple representations of the Weil-Deligne group of $F$. We give strong upper and lower bounds for the Artin and Swan exponents of…

In this note we give a new existence proof for the universal extension classes for $GL_2$ previously constructed by Friedlander and Suslin via the theory of strict polynomial functors. The key tool in our approach is a calculation of Parker…

In $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss's genus theory. In…

Let $\Omega$ be a finite symmetric subset of GL$_n(\mathbb{Z}[1/q_0])$, and $\Gamma:=\langle \Omega \rangle$. Then the family of Cayley graphs $\{{\rm Cay}(\pi_m(\Gamma),\pi_m(\Omega))\}_m$ is a family of expanders as $m$ ranges over fixed…

Working over a field $k$ of characteristic zero, this paper studies algebraic actions of $SL_2(k)$ on affine $k$-domains by defining and investigating fundamental pairs of derivations. There are three main results: (1) The Structure Theorem…

We determine the asymptotic growth of extensions of local function fields of characteristic p counted by discriminant, where the Galois group is a subgroup of the affine group AGL_1(p). More general, we solve the corresponding counting…

The main results in this thesis deal with the representation growth of certain classes of groups. In chapter $1$ we present the required preliminary theory. In chapter $2$ we introduce the Congruence Subgroup Problem for an algebraic group…

A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois…

We consider the low-energy effective action for the Coulomb phase of an $N = 2$ supersymmetric gauge theory with a rank one gauge group. The $N = 2$ superspace formalism is naturally invariant under an $SL(2, {\bf Z})$ group of duality…

We prove that torsion in the abelianizations of open normal subgroups in finitely presented pro-$p$ groups can grow arbitrarily fast. By way of contrast in $\mathbb Z_p$- analytic groups the torsion growth is at most polynomial.

In previous work, the authors established various bounds for the dimensions of degree $n$ cohomology and $\Ext$-groups, for irreducible modules of semisimple algebraic groups $G$ (in positive characteristic $p$) and (Lusztig) quantum groups…