Related papers: Scott's formula and Hurwitz groups

The genus spectrum of a finite group $G$ is the set of all $g\geq 2$ such that $G$ acts faithfully and orientation-preserving on a closed compact orientable surface of genus $g$. This article is an overview of some results relating the…

We develop theorems which produce a multitude of hyperbolic triples for the finite classical groups. We apply these theorems to prove that every quasisimple group except Alt(5) and SL_2(5) is a Beauville group. In particular, we settle a…

Let $\F_q$ be the finite field with $q$ elements, $p=\Char \F_q$. The group $\GL_2(\F_q)$ acts naturally in the set of irreducible polynomials over $\F_q$ of degree at least $2$. In this paper we are interested in the characterization and…

We investigate a (potentially infinite) series of subfactors, called $3^n$ subfactors, including $A_4$, $A_7$, and the Haagerup subfactor as the first three members corresponding to $n=1,2,3$. Generalizing our previous work for odd $n$, we…

We prove that the exceptional group $E_6(2)$ is not a Hurwitz group. In the course of proving this, we complete the classification up to conjugacy of all Hurwitz subgroups of $E_6(2)$, in particular, those isomorphic to $L_2(8)$ and…

We prove a factorization theorem for Fuchsian groups similar to those proved by Agol and Liu for 3-manifold groups. As an application, we build Makanin-Razborov diagrams, which parametrize the collection of all discrete representations from…

We investigate a possible connection between the $FSZ$ properties of a group and its Sylow subgroups. We show that the simple groups $G_2(5)$ and $S_6(5)$, as well as all sporadic simple groups with order divisible by $5^6$ are not $FSZ$,…

The prime graph of a finite group $G$ is denoted by $\ga(G)$ whose vertex set is $\pi(G)$ and two distinct primes $p$ and $q$ are adjacent in $\ga(G)$, whenever $G$ contains an element with order $pq$. We say that $G$ is unrecognizable by…

Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\cal L}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$-central in $G$…

We provide a new condition for an absolutely almost simple algebraic group to have good reduction with respect to a discrete valuation of the base field which is formulated in terms of the existence of maximal tori with special properties.…

We reduce a case of the hidden subgroup problem (HSP) in SL(2; q), PSL(2; q), and PGL(2; q), three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group AGL(1; q). These groups act on projective…

In this report we summarize this work, all finite simple groups $G$ can determined uniformly using their orders $|G|$ and the set $\pi_e(G)$ of their element orders.

We study the class of all algebras that are isotopic to a Hurwitz algebra. Isomorphism classes of such algebras are shown to correspond to orbits of a certain group action. A complete, geometrically intuitive description of the category of…

We study the finitely generated Hausdorff spectrum of spinal automorphism groups acting on rooted trees. Given any $\alpha \in [0,1]$, we construct a branch group $G_\alpha$ such that $G_\alpha$ has a finitely generated subgroup $H$ where…

We study the distribution of products of conjugacy classes in finite simple groups, obtaining various effective uniformity results, which give rise to an approximation to a conjecture of Thompson. Our results, combined with work of Gowers…

This manuscript studies a special case of the Hurwitz enumeration problem: for branched covers from genus g compact Riemann surface to the Riemann sphere, with three branch points, and require the branching data at one of the branch points…

In this paper, we determine the finite groups with a Sylow $r$-subgroup contained in a unique maximal subgroup. The proof involves a reduction to almost simple groups, and our main theorem extends earlier work of Aschbacher in the special…

E. Kani has shown that the Hurwitz functor, which parametrizes the (normalized) genus 2 covers of degree 3 of an elliptic curve, is representable. In this paper the corresponding moduli scheme and the universal family are explicitly…

We consider the automorphism groups of various Lorentzian lattices over the Eisenstein, Gaussian, and Hurwitz integers, and in some of them we find reflection groups of finite index. These provide new finite-covolume reflection groups…

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G, then for every subset B of G with $|B| > |G| / k^{1/3}$ we have B^3 =…