arXiv subjects

Pavel Gvozdevsky

Publications and source records attributed to Pavel Gvozdevsky.

14 recordsLinked to original sources

arXiv2025

Abstract isomorphisms of isotropic root graded groups over rings

The celebrated Borel--Tits theorem provides a classification of abstract isomorphisms between (simple) isotropic groups over fields, showing that such isomorphisms arise from field isomorphisms and group-scheme isomorphisms. In this work, we extend the scope of this classification to certain class of group schemes over arbitrary commutative rings. Specifically, we prove that under suitable conditions abstract isomorphisms between the groups of points of isotropic, absolutely simple, adjoint group schemes over rings admit a description analogous to that in the classical setting: namely, they are induced by isomorphisms of ground rings and isomorphisms of the underlying group schemes. This result generalizes the classical theory to a far broader algebraic context and confirms that the rigidity phenomena observed over fields persist over rings.

math.GR↗

arXiv2024

Verbal width in arithmetic Chevalley groups

We prove that the width of any word in a simply connected Chevalley group of rank at least 2 over the ring that is a localisation of the ring of integers in a number field is bounded by a constant that depends only on the root system and on the degree of the number field.

math.GR↗

arXiv2023

Regular bi-interpretability and finite axiomatizability of Chevalley groups

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 system $G_2$ also $1/3$. Under these assumptions we prove that the central quotients of Chevalley groups are regularly bi-interpretable with the corresponding rings, the class of all central quotients of Chevalley groups of a given type is elementarily definable and even finitely axiomatizable (see Definition~2.2). The same holds for adjoint Chevalley groups and for bondedly generated Chevalley groups. We also give an example of Chevalley group with infinite center, which is not bi-interpretable with the corresponding ring and is elementarily equivalent to a group that is not a Chevalley group itself.

math.GR↗

arXiv2023

On countable isotypic structures

We obtain several results concerning the concept of isotypic structures. Namely we prove that any field of finite transcendence degree over a prime subfield is defined by types; then we construct isotypic but not isomorphic structures with countable underlying sets: totally ordered sets, fields, and groups. This answers an old question by B. Plotkin for groups.

math.LO↗

arXiv2023

Bounded reduction for Chevalley groups of types $E_6$ and $E_7$

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 elementary root elements. The bound is given explicitly. This result is an effective version of the early stabilisation of the corresponding $K_1$-functor. We also give part of the proof of similar hypothesis for $E_8$.

math.GR↗

arXiv2022

Overgroups of subsystem subgroups in exceptional groups: inside a sandwich

The current paper is an addition to the previous paper by author, where the overgroup lattice of the elementary subsystem subgroup $E(Δ,R)$ of the Chevalley group $G(Φ,R)$ for a large enough root subsystem $Δ$ was studied. Now we study the connection between the elementary subgroup $\hat{E}(σ)$ given by the net of ideals of the ring $R$ and the stabilizer $S(σ)$ of the corresponding Lie subalgebra of the Chevalley algebra. In particular, we prove that under a certain condition the subgroup $\hat{E}(σ)$ is normal in $S(σ)$, and we also study some properties of the corresponding quotient group.

math.GR↗

arXiv2022

Twisted forms of commutative monoid structures on affine spaces

In this paper, we study affine commutative algebraic monoid structures on affine spaces over an arbitrary field of characteristic zero. We obtain full classification of such structures on $\mathbb{A}_K^2$ and $\mathbb{A}_K^3$ and describe some generalizations on $\mathbb{A}_K^n$ for any dimension $n$.

math.AG↗

arXiv2022

Width of SL(n,O_S,I)

We give an estimate for the width of the congruence subgroup $\mathrm{SL}(n,O_S,I)$ in Tits--Vaserstein generators, where $O_S$ is a localisation of the ring of integers in a number field $K$. We assume that either $K$ has a real embedding, or the ideal $I$ is prime to the number of roots of unity in $K$.

math.GR↗

arXiv2022

Bounded reduction of orthogonal matrices over polynomial rings

We prove that a matrix from the split orthogonal group over a polynomial ring with coefficients in a small-dimensional ring can be reduced to a smaller matrix by a bounded number of elementary orthogonal transformations. The bound is given explicitly. This result is an effective version of the early stabilisation of the orthogonal K1 functor proven by Suslin and Kopeiko. Since the similar effective results for special linear and symplectic groups were obtained by Vaserstein, the present paper closes the problem for split classical groups.

math.GR↗

arXiv2021

Overgroups of subsystem subgroups in exceptional groups: nonideal levels

In the present paper, we practicaly complete the solution of the problem on the description of overgroups of the subsystem subgroup $E(\Delta,R)$ in the Chevalley group $G(\Phi,R)$ over the ring $R$, where $\Phi$ is a simply laced root system, and $\Delta$ is its large enough subsystem. Namely we define objects called levels, and show that for any such an overgroup $H$ there exists a unique level $\sigma$ such that $E(\sigma)\le H\le \mathrm{Stab}_{G(\Phi,R)}(L_{\max}(\sigma))$, where $E(\sigma)$ is an elementary subgroup defined by the level $\sigma$, and $L_{\max}(\sigma)$ is the corresponding Lie subalgebra in the Chevalley algebra. Unlike all the previous papers, now levels can be more complicated objects that the nets of ideals.

math.GR↗

arXiv2021

Overgroups of Levi subgroups I. The case of abelian unipotent radical

In the present paper we prove sandwich classification for the overgroups of the subsystem subgroup $E(Δ,R)$ of the Chevalley group $G(Φ,R)$ for the three types of pair $(Φ,Δ)$ (the root system and its subsystem) such that the group $G(Δ,R)$ is (up to torus) a Levi subgroup of the parabolic subgroup with abelian unipotent radical. Namely we show that for any such an overgroup $H$ there exists a unique pair of ideals $σ$ of the ring $R$ such that $E(Φ,Δ,R,σ)\le H\le N_{G(Φ,R)}(E(Φ,Δ,R,σ))$.

math.GR↗

arXiv2020

Commutator lengths in general linear group over a skew-field

We give an upper and lower estimate for the maximal commutator length of a noncentral element of the elementary subgroup of the general linear group over a skew-field based on the maximal commutator length of an element of the multiplicative group of that skew-field.

math.GR↗

arXiv2019

Improved $K_1$-stability for the embedding $D_5$ into $E_6$

This paper is dedicated to the surjective stability of the $K_1$-functor for Chevalley groups for the embedding $D_5$ into $E_6$. This case was already studed by Plotkin. In the present paper, we improve his result by showing that surjective stability holds under a weaker assumption on a ring. Another result of the present paper shows how the $K_1$-stability can help to study overgroups of subsystem subgroups.

math.GR↗

arXiv2019

Overgroups of subsystem subgroups in exceptional groups: 2A1-proof

In the present paper we prove a weak form of sandwich classification for the overgroups of the subsystem subgroup $E(\Delta,R)$ of the Chevalley group $G(\Phi,R)$ where $\Phi$ is a simply laced root sysetem and $\Delta$ is its sufficiently large subsystem. Namely we show that for any such an overgroup $H$ there exists a unique net of ideals $\sigma$ of the ring $R$ such that $E(\Phi,\Delta,R,\sigma)\le H\le {\mathop{\mathrm{Stab}}\nolimits}_{G(\Phi,R)}(L(\sigma))$ where $E(\Phi,\Delta,R,\sigma)$ is an elementary subgroup associated with the net and $L(\sigma)$ is a corresponding subalgebra of the Chevalley Lie algebra.

math.GR↗