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Yaroslav Vorobets

Publications and source records attributed to Yaroslav Vorobets.

15 recordsLinked to original sources
arXiv2026

Thue-Morse sequence and groups of intermediate growth

We consider the substitution subshift generated by the Thue-Morse substitution $0\to01$, $1\to10$. We prove that the topological full group of the subshift contains a subgroup of intermediate growth. Namely, one group from the family known as the Grigorchuk groups embeds into this group. To obtain our main result, we prove an embedding theorem for topological full groups, and also develop a technique to prove isomorphism of groups using the Schreier graphs.

math.GR↗
arXiv2024

On maximal subgroups of ample groups

The paper is concerned with maximal subgroups of the ample (better known as topological full) groups of homeomorphisms of totally disconnected compact metrizable topological spaces. We describe all maximal subgroups that are stabilizers of finite sets. Under certain assumptions on the ample group (including minimality), we describe all maximal subgroups that are stabilizers of closed sets or stabilizers of partitions into clopen sets. In particular, our results apply to the ample groups associated with Cantor minimal systems and to some Higman-Thompson groups.

math.GR↗
arXiv2021

On Mealy-Moore coding and images of Markov measures

We study the images of the Markov measures under transformations generated by the Mealy automata. We find conditions under which the image measure is absolutely continuous or singular relative to the Markov measure. Also, we determine statistical properties of the image of a generic sequence.

math.DS↗
arXiv2020

On the topological full group containing the Grigorchuk group

We consider the topological full group of a substitution subshift induced by a substitution $a\to aca$, $b\to d$, $c\to b$, $d\to c$. This group is interesting since the Grigorchuk group naturally embeds into it. We show that the topological full group is finitely generated and give a simple generating set for it.

math.GR↗
arXiv2018

Automatic Logarithm and Associated Measures

We introduce the notion of the Automatic Logarithm $\mathcal L_{\mathcal A, \mathcal B}$ with the purpose of studying the expanding properties of Schreier graphs of action of the group generated by two finite initial Mealy automata $\mathcal A$ and $\mathcal B$ on the levels of a regular $d$-ary rooted tree $\mathcal T$, where $\mathcal A$ is level-transitive and of bounded activity. $\mathcal L_{\mathcal A, \mathcal B}$ computes the lengths of chords in this family of graphs. Formally, $\mathcal L$ is a map $\partial \mathcal T \rightarrow \mathbb{Z}_d$ from the boundary of the tree to the integer $p$-adics whose values are determined by a Moore machine. The distribution of its outputs yields a probabilistic measure $μ$ on $\partial \mathcal T$, which in some cases can be computed by a Mealy-type machine (we then say that $μ$ is finite-state). We provide a criterion to determine whether $μ$ is finite-state. A number of examples illustrating the different cases with $\mathcal A$ being the adding machine is provided.

math.GR↗
arXiv2013

On growth of random groups of intermediate growth

We study the growth of typical groups from the family of $p$-groups of intermediate growth constructed by the second author. We find that, in the sense of category, a generic group exhibits oscillating growth with no universal upper bound. At the same time, from a measure-theoretic point of view (i.e., almost surely relative to an appropriately chosen probability measure), the growth function is bounded by $e^{n^α}$ for some $α<1$.

math.GR↗
arXiv2011

Notes on the Schreier graphs of the Grigorchuk group

The paper is concerned with the space of the marked Schreier graphs of the Grigorchuk group and the action of the group on this space. In particular, we describe an invariant set of the Schreier graphs corresponding to the action on the boundary of the binary rooted tree and dynamics of the group action restricted to this invariant set.

math.DS↗
arXiv2009

On a substitution subshift related to the Grigorchuk group

We study dynamics of a substitution subshift given by the substitution a->aca, b->d, c->b, d->c that is related to the Grigorchuk group. This dynamical system is shown to be, up to a countable set, conjugated to the binary odometer.

math.DS↗
arXiv2009

Fermat's spiral and the line between Yin and Yang

Let $D$ denote a disk of unit area. We call a subset $A$ of $D$ perfect if it has measure 1/2 and, with respect to any axial symmetry of $D$, the maximal symmetric subset of $A$ has measure 1/4. We call a curve $β$ in $D$ an yin-yang line if $β$ splits $D$ into two congruent perfect sets, $β$ crosses each concentric circle of $D$ twice, $β$ crosses each radius of $D$ once. We prove that Fermat's spiral is a unique yin-yang line in the class of smooth curves algebraic in polar coordinates.

math.CO↗
arXiv2006

Automata over a binary alphabet generating free groups of even rank

We construct automata over a binary alphabet with $2n$ states, $n\geq 2$, whose states freely generate a free group of rank $2n$. Combined with previous work, this shows that a free group of every finite rank can be generated by finite automata over a binary alphabet. We also construct free products of cyclic groups of order two via such automata.

math.GR↗
arXiv2006

On a series of finite automata defining free transformation groups

We introduce two series of finite automata starting from the so-called Aleshin and Bellaterra automata. We prove that each automaton in the first series defines a free non-Abelian group while each automaton in the second series defines the free product of groups of order 2. Furthermore, these properties are shared by disjoint unions of any number of distinct automata from either series.

math.GR↗
arXiv2003

Periodic geodesics on translation surfaces

As shown by Masur in 80s, for any translation surface there exists a periodic geodesic of bounded length, the directions of periodic geodesics are dense in the unit circle, and the number of cylinders of periodic geodesics of length at most T admits upper and lower quadratic estimates. The main goal of this paper is to generalize the above results, with special interest in obtaining effective estimates. In addition, we apply results of Eskin and Masur to establish some properties of periodic geodesics on generic translation surfaces.

math.DS↗