About

Rostislav Grigorchuk is a Distinguished Professor at Texas A&M University in the College of Arts and Sciences. His research areas include Algebra & Combinatorics, Groups & Dynamics, with research interests specifically focused on Combinatorial Group Theory. He holds a habilitation from the Steklov Mathematical Institute obtained in 1985, a Ph.D. from Moscow Lomonosov State University earned in 1978, and a B.S. and M.S. from the same university completed in 1975. His academic and research career is centered on advanced mathematical theories and group dynamics, contributing significantly to his field.

Research topics

• Quantum mechanics
• Pure mathematics
• Mathematics
• Physics
• Mathematical analysis

Selected publications

• Directional counting for regular languages

  Algebra and Discrete Mathematics · 2026-01-01

  preprintOpen access1st authorCorresponding

  We explain how certain tools from convex analysis and probability theory may be used in order to obtain counting results for the number of words with prescribed frequencies of letters in regular languages.

  PublisherOA PDFDOI

• Multivariate growth and cogrowth

  Carpathian Mathematical Publications · 2025-05-21

  articleOpen access1st authorCorresponding

  We deal with a multivariate growth series $\Gamma_L(\mathbf{z})$, $\mathbf{z} \in \mathbb{C}^d$, associated with a regular language $L$ over an alphabet of cardinality $d \geq 2$. Our focus is on languages coming from subgroups of the free group $F_m$ of finite rank $m$ and from the subshifts of finite type. We suggest a tool for computing the rate of growth $\varphi_L(\mathbf{r})$ of $L$ in the direction $\mathbf{r} \in \mathbb{R}^d$. Using the concave growth condition introduced by the second author in [Comment. Math. Helv. 2002, 77 (3), 563-608] and the results of Convex Analysis we represent $\psi_L(\mathbf{r}) = \log\left(\varphi_L(\mathbf{r})\right)$ as a support function of a convex set that is the closure of the $\textrm{Relog}$ image of the domain of absolute convergence of $\Gamma_L(\mathbf{z})$. This allows us to compute $\psi_L(\mathbf{r})$ in some cases, including a Fibonacci language or a language of freely reduced words representing elements of a free group $F_2$. Also we show that the methods of the Large Deviation Theory can be used as an alternative approach.

  PublisherOA PDFDOI

• Liftable self-similar groups and scale groups

  Transactions of the American Mathematical Society · 2025-10-01

  article1st authorCorresponding

  We canonically identify the groups of isometries and dilations of local fields and their rings of integers with subgroups of the automorphism group of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis d plus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(d+1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -regular tree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T overTilde Subscript d plus 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>T</mml:mi> <mml:mo> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\widetilde T_{d+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the residual degree. Then we introduce the class of liftable self-similar groups acting on a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -regular rooted tree whose ascending HNN extensions act faithfully and vertex transitively on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T overTilde Subscript d plus 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>T</mml:mi> <mml:mo> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\widetilde T_{d+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> fixing one of the ends. The closures of these extensions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A u t left-parenthesis upper T overTilde Subscript d plus 1 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>T</mml:mi> <mml:mo> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Aut(\widetilde T_{d+1})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are totally disconnected locally compact groups that belong to the class of scale groups as defined by Willis [Scale groups, 2022]. We give numerous examples of liftable groups coming from self-similar groups acting essentially freely on the boundaries of rooted trees or groups admitting finite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -presentations. In particular, we show that the finitely presented group constructed by the first author [Mat. Sb. 189 (1998), pp. 79–100] and the finitely presented HNN extension of the Basilica group constructed by Grigorchuk and Żuk [Spectral properties of a torsion-free weakly branch group defined by a three state automaton, Amer. Math. Soc., Providence, RI, 2002, pp. 57–82] embed into the group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D left-parenthesis double-struck upper Q 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">D</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal D(\mathbb {Q}_2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of dilations of the field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {Q}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic numbers. These actions, translated to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://

  PublisherDOI

• On the structure of finitely generated subgroups of branch groups

  Journal of Combinatorial Algebra · 2025-05-14 · 1 citations

  articleOpen access

  Motivated by the study of profinite topology in branch groups, we prove a structural result about their finitely generated subgroups. More precisely, we show that finitely generated subgroups of a branch group with the subgroup induction property have a block structure, which roughly means that, up to a finite index, they are products of finite index subgroups, embedded in the group in a way that is coherent with its branch action on the rooted tree.

  PublisherDOI

• On the Second 2-Dimensional Rational Map Associated with the First Group of Intermediate Growth

  Bukovinian Mathematical Journal · 2025-12-24

  articleOpen access1st authorCorresponding

  We show that the second rational map $G$ associated with the group $\mathcal{G}$ of intermediate growth constructed by the first author in 1980 is semiconjugate with the antiholomorphic map $z\rightarrow\bar{z}^2$. For doing this we use a family of $G$--invariant curves found by the third author and for each invariant curve, create a Markov partition of it.

  PublisherOA PDFDOI

• On maximal subgroups of ample groups

  arXiv (Cornell University) · 2024-03-25

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

• Branch actions and the structure lattice

  Algebra and Discrete Mathematics · 2024-01-01

  articleOpen accessSenior author

  J. S. Wilson proved in 1971 an isomorphism between the structural lattice associated to a group belonging to his second class of groups with every proper quotient finite and the Boolean algebra of clopen subsets of Cantor’s ternary set. In this paper we generalize this isomorphism to the class of branch groups. Moreover, we show that for every faithful branch action of a group \(G\) on a spherically homogeneous rooted tree \(T\) there is a canonical \(G\)-equivariant isomorphism between the Boolean algebra associated to the structure lattice of \(G\) and the Boolean algebra of clopen subsets of the boundary of \(T\) .

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• On the structure of finitely generated subgroups of branch groups

  arXiv (Cornell University) · 2024-02-23

  preprintOpen access

  We describe the block structure of finitely generated subgroups of branch groups with the so-called subgroup induction property, including the first Grigorchuk group $\mathcal{G}$ and the torsion GGS groups.

  PublisherOA PDFDOI

• Laplace and Schrödinger operators without eigenvalues on homogeneous amenable graphs

  Asian Journal of Mathematics · 2024-01-01

  article1st authorCorresponding

  A one-by-one exhaustion is a combinatorial/geometric condition which excludes eigenvalues from the spectra of Laplace and Schrodinger operators on graphs. Isoperimetric inequalities in graphs with a cocompact automorphism group provide an upper bound on the von Neumann dimension of the space of eigenfunctions. Any finitely generated indicable amenable group has a Cayley graph without eigenvalues. There exists a finitely generated group G with finite generating sets S and S' such that the adjacency operator of the Cayley graph of (G,S) has no eigenvalue while the adjacency operator of the Cayley graph of (G,S') has pure point spectrum.

  PublisherDOI

• Laplace and Schr&amp;ouml;dinger operators without eigenvalues on homogeneous amenable graphs

  Asian Journal of Mathematics · 2024-01-01

  article1st authorCorresponding
  PublisherDOI

Recent grants

• Groups of intermediate growth

  NSF · $275k · 2012–2015

• Algebraic, combinatorial, spectral and algorithmic properties of groups generated by finite automata

  NSF · $192k · 2006–2009

Frequent coauthors

• Tatiana Nagnibeda

  27 shared

• Rostyslav Kravchenko

  23 shared

• Laurent Bartholdi

  21 shared

• Dmytro Savchuk

  18 shared

• Artem Dudko

  18 shared

• Volodymyr Nekrashevych

  Texas A&M University

  18 shared

• Mustafa Gökhan Benli̇

  Middle East Technical University

  16 shared

• Pierre de la Harpe

  University of Geneva

  15 shared

Education

• PhD, Department of Mechanics and Mathematics

  Lomonosov Moscow State University

  1978