Related papers: A Pansiot-type subword complexity theorem for auto…

We investigate the homology representation of the symmetric group on rank-selected subposets of subword order. We show that the homology module for words of bounded length, over an alphabet of size $n,$ decomposes into a sum of tensor…

By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate…

Following Plotkin we say that the automorphism $x$ of the group $G$ is a nil-automorphism if, for every $g\in G$, there exists $n=n(g)$ such that $[g,_n x]=1$. If the integer $n$ can be chosen independently of $g$, then $x$ is said to be…

We show that the automorphism group of affine n-space $A^n$ determines $A^n$ up to isomorphism: If $X$ is a connected affine variety such that $Aut(X)$ is isomorphic to $Aut(A^n)$ as ind-groups, then $X$ is isomorphic to $A^n$ as a variety.…

Let $M(A,I)$ be a free partially commutative monoid with involution and $G(A,I)$ be its quotient group, e.g. a right-angled Artin or Coxeter group. Given a system of word equations over $M(A,I)$ with recognizable constraints with input size…

We discuss the following question of G. Makanin from ``Kourovka notebook'': does there exist an algorithm to determine is for an arbitrary pair of words $U$ and $V$ of a free group $F_n$ and an arbitrary automorphism $\phi \in Aut(F_n)$ the…

We prove two results about the natural representation of a group G of automorphisms of a normal projective threefold X on its second cohomology. We show that if X is minimal then G, modulo a normal subgroup of null entropy, is embedded as a…

Let $A$ be an $n \times n$ matrix, $X$ be an $n \times p$ matrix and $Y = AX$. A challenging and important problem in data analysis, motivated by dictionary learning and other practical problems, is to recover both $A$ and $X$, given $Y$.…

We prove that, for a finitely generated residually finite group, having solvable word problem is not a sufficient condition to be a subgroup of a finitely presented residually finite group. The obstruction is given by a residually finite…

Let $FH$ be a Frobenius group with kernel $F$ and complement $H$, acting coprimely on the finite solvable group $G$ by automorphisms. We prove that if $C_{G}(H)$ is of Fitting length $n$ then the index of the $n$-th Fitting subgroup…

The article focuses on word (or string) attractors, which are sets of positions related to the text compression efficiency of the underlying word. The article presents two combinatorial algorithms based on Suffix automata or Directed…

Consider a group word w in n letters. For a compact group G, w induces a map G^n \rightarrow G$ and thus a pushforward measure {\mu}_w on G from the Haar measure on G^n. We associate to each word w a 2-dimensional cell complex X(w) and…

Let G be a group of automorphisms of a compact K\"ahler manifold X of dimension n and N(G) the subset of null-entropy elements. Suppose G admits no non-abelian free subgroup. Improving the known Tits alternative, we obtain that, up to…

For a finitely generated free group F_n, of rank at least 2, any finite subgroup of Out(F_n) can be realized as a group of automorphisms of a graph with fundamental group F_n. This result, known as Out(F_n) realization, was proved by…

We study the interaction between the group law on an abelian variety and the additive structure induced on its image under a morphism to projective space. Let $A/F$ be a simple abelian variety, $f:A \rightarrow \mathbb{P}^n$ be a morphism…

A theorem of A.A. Brudno says that the Kolmogorov-Sinai entropy of a subshift X over $\mathbb{N}$ with respect to an ergodic measure $\mu$ equals the asymptotic Kolmogorov complexity of almost every word $\omega$ in X. The purpose of this…

In this paper we describe the structure of a group of conjugating automorphisms $C_n$ of free group and prove that this structure is similar to the structure of a braid group $B_n$ with $n>1$ strings. We find the linear representation of…

The fundamental group of a finite graph of groups with trivial edge groups is a free product. We are interested in those outer automorphisms of such a free product that permute the conjugacy classes of the vertex groups. We show that in…

Consider a smooth connected algebraic group $G$ acting on a normal projective variety $X$ with an open dense orbit. We show that Aut($X$) is a linear algebraic group if so is $G$; for an arbitrary $G$, the group of components of Aut($X$) is…

We prove that an arbitrary compact metrizable group can be realized as the automorphism group of a graphing; this is a continuous analogue to Frucht's theorem recovering arbitrary finite groups are automorphism groups of finite graphs. The…