Related papers: Verbal width in arithmetic Chevalley groups
An estimate on the commutator width is given for Chevalley groups over rings of stable rank 1, and the general method suitable for other rings of small dimension.
We prove that bi-invariant word metrics are bounded on certain Chevalley groups. As an application we provide restrictions on Hamiltonian actions of such groups.
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…
We prove that finite index subgroups in S-arithmetic Chevalley groups are bounded.
We construct words with small image in a given finite alternating or unimodular group. This shows that word width in these groups is unbounded in general.
In this paper we establish a definitive result which almost completely closes the problem of bounded elementary generation for Chevalley groups of rank $\ge 2$ over arbitrary Dedekind rings $R$ of arithmetic type, with uniform bounds.…
We prove two results about width of words in $SL_n(\mathbb{Z})$. The first is that, for every $n \geq 3$, there is a constant $C(n)$ such that the width of any word in $SL_n(\mathbb{Z})$ is less than $C(n)$. The second result is that, for…
We prove that an element from the Chevalley group of type $E_6$ or $E_7$ over a polynomial ring with coefficients in a small-dimensional ring can be reduced to an element of certain proper subsystem subgroup by a bounded number of…
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…
For finite reflection groups of types A and B, we determine the diameter of the graph whose vertices are reduced words for the longest element and whose edges are braid relations. This is deduced from a more general theorem that applies to…
In the current article we study structure of a Chevalley group $G(R)$ over a commutative ring $R$. We generalize and improve the following results: (1) standard, relative, and multi-relative commutator formulas; (2) nilpotent structure of…
We formulate and prove relative versions of several classical decompositions known in the theory of Chevalley groups over commutative rings. As an application we obtain upper estimates for the width of principal congruence subgroups in…
An adjoint Chevalley group of rank at least 2 over a rational algebra (or a similar ring), its elementary subgroup, and the corresponding Lie ring have the same automorphism group. These automorphisms are explicitly described.
We show that for Chevalley groups G(R) of rank at least 2 over a ring R the root subgroups are essentially (nearly always) the double centralizers of corresponding root elements. In very many cases this implies that R and G(R) are…
In this paper we prove that every automorphism of a Chevalley group (or its elementary subgroup) with root system of rank >1 over a commutative ring (with 1/2 for the systems A_2, F_4, B_l, C_l; with 1/2 and 1/3 for the system G_2) is…
A subset of a group is characteristic if it is invariant under every automorphism of the group. We study word length in fundamental groups of closed hyperbolic surfaces with respect to characteristic generating sets consisting of a finite…
We prove that Chevalley groups over polynomial rings $\mathbb F_q[t]$ and over Laurent polynomial $\mathbb F_q[t,t^{-1}]$ rings, where $\mathbb F_q$ is a finite field, are boundedly elementarily generated. Using this we produce explicit…
In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group of degree n, the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for…
In this paper we consider Chevalley groups over commutative rings with~$1$, constructed by irreducible root systems of rank $>1$. We always suppose that for the systems $A_2, B_\ell, C_\ell, F_4, G_2$ our rings contain $1/2$ and for the…
We prove that every locally inner endomorphism of a Chevalley group (or its elementary subgroup) over a local ring with an irreducible root system of rank >1 (with 1/2 for the systems A_2, F_4, B_l, C_l and with 1/3 for the system G_2) is…