Related papers: The dichotomy property of ${\rm SL}_2(R)$-A short …

Let $A$ be a commutative ring, and assume every non-trivial ideal of $A$ has finite-index. We show that if ${\rm{SL}}_n(A)$ has bounded elementary generation then every conjugation-invariant norm on it is either discrete or precompact. If…

A group is called strongly bounded, if the speed with which it is generated by finitely many conjugacy classes has a positive, lower bound only dependent on the number of the conjugacy classes in question rather than the actual conjugacy…

This paper is concerned with the diameter of certain word norms on S-arithmetic split Chevalley groups. Such groups are well known to be boundedly generated by root elements. We prove that word metrics given by conjugacy classes on…

This paper gives an explicit argument to show strong boundedness for ${\rm Sp}_{2n}(R)$ for $R$ a ring of S-algebraic integers or a semi-local ring. This gives a quantitative version of a related abstract result in a previous paper of the…

From the cohomological point of view the symplectomorphism group $Sympl (M)$ of a symplectic manifold is `` tamer'' than the diffeomorphism group. The existence of invariant polynomials in the Lie algebra $\frak {sympl }(M)$, the symplectic…

For a finite group acting on a polynomial ring, the Chevalley-Shephard-Todd Theorem proves that the fixed subring is isomorphic to a polynomial ring if and only if the group is generated by pseudo-reflections. In recent years, progress was…

In this paper, we prove two structure theorems for twisted Chevalley groups $G_\sigma (R)$ over a commutative ring $R$ with unity. The first theorem concerns the normality of $E'_\sigma (R,J)$, the elementary congruence subgroups at level…

Let G be a Chevalley group scheme and B<=G a Borel subgroup scheme, both defined over Z. Let K be a global function field, S be a finite non-empty set of places over K, and O_S be the corresponding S-arithmetic ring. Then, the S-arithmetic…

Let R be a Dedekind domain, and let G be a simply connected Chevalley-Demazure group scheme of rank =>2. We prove that G(R[x_1,...,x_n])=G(R)E(R[x_1,...,x_n]) for any n=>1. This extends the corresponding results of A. Suslin and F.…

We construct the finite-dimensional continuous complex representations of $\mathrm{SL}_2$ over compact discrete valuation rings of even residual characteristic. We also prove that the complex group algebras of $\mathrm{SL}_2$ over finite…

In this series of papers, we investigate properties of a finite group which are determined by its low degree irreducible representations over a number field $F$, i.e. its representations on matrix rings $\operatorname{M}_n(D)$ with $n \leq…

In these notes we determine the finiteness length of the groups G(O_S) where G is an F_q-isotropic, connected, noncommutative, almost simple F_q-group and O_S is one of F_q[t], F_q[t^{-1}], and F_q[t,t^{-1}]. That is, k = F_q(t) and S…

Chevalley's theorem and it's converse, the Sheppard-Todd theorem, assert that finite reflection groups are distinguished by the fact that the ring of invariant polynomials is freely generated. We show that in the Euclidean case, a weaker…

Recently there has been a lot of research and progress in profinite groups. We survey some of the new results and discuss open problems. A central theme is decompositions of finite groups into bounded products of subsets of various kinds…

Let $G$ be a simply connected Chevalley--Demazure group scheme without $SL_2$-factors. For any unital commutative ring $R$, we denote by $E(R)$ the standard elementary subgroup of $G(R)$, that is, the subgroup generated by the elementary…

Chevalley's theorem states that for any simple finite dimensional Lie algebra G (1) the restriction homomorphism of the algebra of polynomials on G onto the Cartan subalgebra H induces an isomorphism between the algebra of G-invariant…

Let W be a finite reflection group acting orthogonally on R^n, P be the Chevalley polynomial mapping determined by an integrity basis of the algebra of W-invariant polynomials, and h be the highest degree of the coordinate polynomials in…

The $SL_2(\mathbb{C})$ character variety $X(M)$ has emerged as an important tool in studying the topology of hyperbolic 3-manifolds. Chinburg-Reid-Stover constructed arithmetic invariants stemming from a canonical Azumaya algebra over the…

We present new Hopf algebras with the dual Chevalley property by determining all semisimple Hopf algebras Morita-equivalent to a group algebra over a finite group, for a list of groups supporting a non-trivial finite-dimensional Nichols…

We define a GL-variety to be a (typically infinite dimensional) algebraic variety equipped with an action of the infinite general linear group under which the coordinate ring forms a polynomial representation. Such varieties have been used…