Rostislav Grigorchuk

Rostislav Ivanovich Grigorchuk (Ukrainian: Ростисла́в Iва́нович Григорчу́к; b. February 23, 1953) is a mathematician working in different areas of mathematics including group theory, dynamical systems, geometry and computer science. He holds the rank of Distinguished Professor in the Mathematics Department of Texas A&M University. Grigorchuk is particularly well known for having constructed, in a 1984 paper,[1] the first example of a finitely generated group of intermediate growth, thus answering an important problem posed by John Milnor in 1968. This group is now known as the Grigorchuk group[2][3][4][5][6] and it is one of the important objects studied in geometric group theory, particularly in the study of branch groups, automaton groups and iterated monodromy groups. Grigorchuk is one of the pioneers of asymptotic group theory as well as of the theory of dynamically defined groups. He introduced the notion of branch groups[7][8][9][10] and developed the foundations of the related theory. Grigorchuk, together with his collaborators and students, initiated the theory of groups generated by finite Mealy type automata,[11][12][13] interpreted them as groups of fractal type,[14][15] developed the theory of groups acting on rooted trees,[16] and found numerous applications[17][18][19] of these groups in various fields of mathematics including functional analysis, topology, spectral graph theory, dynamical systems and ergodic theory.

Rostislav Ivanovich Grigorchuk
Born (1953-02-23) February 23, 1953
Vyshnivets, Ternopil Oblast, Ukraine
Alma materLomonosov Moscow State University
Known forresearcher in geometric group theory, discovering the Grigorchuk group
AwardsAward of Moscow Mathematical Society (1979), Bogolyubov Prize of National Academy of Sciences of Ukraine (2015), Leroy P. Steele Prize (2015), Humboldt Research Award by Germany’s Alexander von Humboldt Foundation (2020)
Scientific career
FieldsMathematics
InstitutionsTexas A&M University

Biographical data

Grigorchuk was born on February 23, 1953, in Ternopil Oblast, now Ukraine (in 1953 part of the USSR).[20] He received his undergraduate degree in 1975 from Moscow State University. He obtained a PhD (Candidate of Science) in Mathematics in 1978, also from Moscow State University, where his thesis advisor was Anatoly M. Stepin. Grigorchuk received a habilitation (Doctor of Science) degree in Mathematics in 1985 at the Steklov Institute of Mathematics in Moscow.[20] During the 1980s and 1990s, Rostislav Grigorchuk held positions at the Moscow State University of Transportation, and subsequently at the Steklov Institute of Mathematics and Moscow State University.[20] In 2002 Grigorchuk joined the faculty of Texas A&M University as a Professor of Mathematics, and he was promoted to the rank of Distinguished Professor in 2008.[21]

Rostislav Grigorchuk gave an invited address at the 1990 International Congress of Mathematicians in Kyoto[22] an AMS Invited Address at the March 2004 meeting of the American Mathematical Society in Athens, Ohio[23] and a plenary talk at the 2004 Winter Meeting of the Canadian Mathematical Society.[24]

Grigorchuk is the Editor-in-Chief of the journal "Groups, Geometry and Dynamics",[25] published by the European Mathematical Society, and is or was a member of the editorial boards of the journals "Mathematical Notes",[26] "International Journal of Algebra and Computation",[27] "Journal of Modern Dynamics",[28] "Geometriae Dedicata",[29] "Ukrainian Mathematical Journal",[30] "Algebra and Discrete Mathematics",[31] "Carpathian Mathematical Publications",[32] "Bukovinian Mathematical Journal",[33] and "Matematychni Studii".[34]

Mathematical contributions

Grigorchuk is most well known for having constructed the first example of a finitely generated group of intermediate growth which now bears his name and is called the Grigorchuk group (sometimes it is also called the first Grigorchuk group since Grigorchuk constructed several other groups that are also commonly studied). This group has growth that is faster than polynomial but slower than exponential. Grigorchuk constructed this group in a 1980 paper[35] and proved that it has intermediate growth in a 1984 article.[1] This result answered a long-standing open problem posed by John Milnor in 1968 about the existence of finitely generated groups of intermediate growth. Grigorchuk's group has a number of other remarkable mathematical properties. It is a finitely generated infinite residually finite 2-group (that is, every element of the group has a finite order which is a power of 2). It is also the first example of a finitely generated group that is amenable but not elementary amenable, thus providing an answer to another long-standing problem, posed by Mahlon Day in 1957.[36] Also Grigorchuk's group is "just infinite": that is, it is infinite but every proper quotient of this group is finite.[2]

Grigorchuk's group is a central object in the study of the so-called branch groups and automata groups. These are finitely generated groups of automorphisms of rooted trees that are given by particularly nice recursive descriptions and that have remarkable self-similar properties. The study of branch, automata and self-similar groups has been particularly active in the 1990s and 2000s and a number of unexpected connections with other areas of mathematics have been discovered there, including dynamical systems, differential geometry, Galois theory, ergodic theory, random walks, fractals, Hecke algebras, bounded cohomology, functional analysis, and others. In particular, many of these self-similar groups arise as iterated monodromy groups of complex polynomials. Important connections have been discovered between the algebraic structure of self-similar groups and the dynamical properties of the polynomials in question, including encoding their Julia sets.[37]

Much of Grigorchuk's work in the 1990s and 2000s has been on developing the theory of branch, automata and self-similar groups and on exploring these connections. For example, Grigorchuk, with co-authors, obtained a counter-example to the conjecture of Michael Atiyah about L2-betti numbers of closed manifolds.[38][39]

Grigorchuk is also known for his contributions to the general theory of random walks on groups and the theory of amenable groups, particularly for obtaining in 1980[40] what is commonly known (see for example[41][42][43]) as Grigorchuk's co-growth criterion of amenability for finitely generated groups.

Awards and honors

In 1979 Rostislav Grigorchuk was awarded the Moscow Mathematical Society.[44]

In 1991 he obtained Fulbright Senior Scholarship,[45] Columbia University, New York.

In 2003 an international group theory conference in honor of Grigorchuk's 50th birthday was held in Gaeta, Italy.[46] Special anniversary issues of the "International Journal of Algebra and Computation",[47] the journal "Algebra and Discrete Mathematics"[20] and the book "Infinite Groups: Geometric, Combinatorial and Dynamical Aspects"[48] were dedicated to Grigorchuk's 50th birthday.

In 2009 Grigorchuk R.I. was awarded the Association of Former Students Distinguished Achievement in Research,[49] Texas A&M University.

In 2012 he became a fellow of the American Mathematical Society.[50]

In 2015 Rostislav Grigorchuk was awarded the AMS Leroy P. Steele Prize for Seminal Contribution to Research.[51] In addition, in this year he became a laureate[52] of Bogolyubov Prize of Ukrainian Academy of Science.

In 2020 Grigorchuk R.I. has been elected as a laureate[53] of the prestigious Humboldt Research Award by Germany’s Alexander von Humboldt Foundation.

See also

References
  1. R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya. vol. 48 (1984), no. 5, pp. 939-985
  2. Pierre de la Harpe. Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago. ISBN 0-226-31719-6
  3. Laurent Bartholdi. The growth of Grigorchuk's torsion group. International Mathematics Research Notices, 1998, no. 20, pp. 1049-1054
  4. Tullio Ceccherini-Silberstein, Antonio Machì, and Fabio Scarabotti. The Grigorchuk group of intermediate growth. Rendiconti del Circolo Matematico di Palermo (2), vol. 50 (2001), no. 1, pp. 67-102
  5. Yu. G. Leonov. On a lower bound for the growth function of the Grigorchuk group. (in Russian). Matematicheskie Zametki, vol. 67 (2000), no. 3, pp. 475-477; translation in: Mathematical Notes, vol. 67 (2000), no. 3-4, pp. 403-405
  6. Roman Muchnik, and Igor Pak. Percolation on Grigorchuk groups. Communications in Algebra, vol. 29 (2001), no. 2, pp. 661-671.
  7. Grigorchuk R. I.Just infinite branch groups. New horizons in pro-p groups.Progr. Math., 184, Birkhäuser Boston, Boston, MA, 2000, 121–179.
  8. Bartholdi, Laurent; Grigorchuk, Rostislav I.; Šuniḱ, Zoran.Branch groups. Handbook of algebra.Vol. 3, 989–1112, Handb. Algebr., 3, Elsevier/North-Holland, Amsterdam, 2003.
  9. Grigorchuk Rostislav.Solved and unsolved problems around one group. Infinite groups: geometric, combinatorial and dynamical aspects.117–218, Progr. Math., 248, Birkhäuser, Basel, 2005.
  10. de la Harpe, Pierre.Topics in geometric group theory. (English summary).Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000.
  11. Grigorchuk R. I.; Nekrashevich, V. V.; Sushchanskiĭ, V. I.Automata, dynamical systems, and groups. (Russian).Tr. Mat. Inst. Steklova 231 (2000), Din. Sist., Avtom. i Beskon. Gruppy, 134–214; translation in Proc. Steklov Inst. Math. 2000, no. 4(231), 128–203.
  12. Bondarenko, Ievgen; Grigorchuk, Rostislav; Kravchenko, Rostyslav; Muntyan, Yevgen; Nekrashevych, Volodymyr; Savchuk, Dmytro; Šunić, Zoran.On classification of groups generated by 3-state automata over a 2-letter alphabet.Algebra Discrete Math. 2008, no. 1, 1–163.
  13. Ceccherini-Silberstein, Tullio and Coornaert, Michel.Cellular automata and groups.Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010.
  14. Bartholdi, Laurent; Grigorchuk, Rostislav; Nekrashevych, Volodymyr.From fractal groups to fractal sets.Fractals in Graz 2001, 25–118, Trends Math., Birkhäuser, Basel, 2003.
  15. Grigorchuk, Rostislav; Nekrashevych, Volodymyr; Šunić, Zoran.From self-similar groups to self-similar sets and spectra. Fractal geometry and stochastics V. 175–207, Progr. Probab., 70, Birkhäuser/Springer, Cham, 2015.
  16. Grigorchuk R. I.Some problems of the dynamics of group actions on rooted trees. (Russian)Tr. Mat. Inst. Steklova 273 (2011).
  17. Grigorchuk, Rostislav; Šunić, Zoran.Self-similarity and branching in group theory.Groups St. Andrews 2005. Vol. 1, 36–95, London Math. Soc. Lecture Note Ser., 339, Cambridge Univ. Press, Cambridge, 2007.
  18. Nekrashevych, Volodymyr.Self-similar groups.Mathematical Surveys and Monographs, 117. American Mathematical Society, Providence, RI, 2005. xii+231 pp
  19. Grigorchuk, Rostislav; Nekrashevych, Volodymyr.Self-similar groups, operator algebras and Schur complement.J. Mod. Dyn. 1 (2007), no. 3, 323–370.
  20. Editorial Statement, Algebra and Discrete Mathematics, (2003), no. 4
  21. 2008 Personal News, Department of Mathematics, Texas A&M University. Accessed January 15, 2010.
  22. R. I. Grigorchuk. On growth in group theory. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pp. 325-338, Math. Soc. Japan, Tokyo, 1991
  23. Spring Central Section Meeting, Athens, OH, March 26-27, 2004. American Mathematical Society. Accessed January 15, 2010.
  24. 2004 Winter Meeting, Canadian Mathematical Society. Accessed January 15, 2010.
  25. Groups, Geometry and Dynamics
  26. Editorial Board, Mathematical Notes
  27. Editorial Board, International Journal of Algebra and Computation
  28. Editorial Board, Journal of Modern Dynamics
  29. Editorial Board, Geometriae Dedicata
  30. Editorial Board, Ukrainian Mathematical Journal
  31. Editorial Board, Algebra and Discrete Mathematics Archived 2008-11-21 at the Wayback Machine
  32. Editorial Board, Carpathian Mathematical Publications
  33. Editorial Board, Bukovinian Mathematical Journal
  34. Editorial Board, Matematychni Studii
  35. R. I. Grigorchuk. On Burnside's problem on periodic groups. (Russian) Funktsionalnyi Analiz i ego Prilozheniya, vol. 14 (1980), no. 1, pp. 53-54
  36. Mahlon M. Day. Amenable semigroups. Illinois Journal of Mathematics, vol. 1 (1957), pp. 509-544.
  37. Volodymyr Nekrashevych. Self-similar groups. Mathematical Surveys and Monographs, 117. American Mathematical Society, Providence, RI, 2005. ISBN 0-8218-3831-8
  38. R. I. Grigorchuk, and A. Zuk. The lamplighter group as a group generated by a 2-state automaton, and its spectrum. Geometriae Dedicata, vol. 87 (2001), no. 1-3, pp. 209--244.
  39. R. I. Grigorchuk, P. Linnell, T. Schick, and A. Zuk. On a question of Atiyah. Comptes Rendus de l'Académie des Sciences, Série I. vol. 331 (2000), no. 9, pp. 663-668.
  40. R. I. Grigorchuk. Symmetrical random walks on discrete groups. Multicomponent random systems, pp. 285-325, Adv. Probab. Related Topics, 6, Marcel Dekker, New York, 1980; ISBN 0-8247-6831-0
  41. R. Ortner, and W. Woess. Non-backtracking random walks and cogrowth of graphs. Canadian Journal of Mathematics, vol. 59 (2007), no. 4, pp. 828-844
  42. Sam Northshield. Quasi-regular graphs, cogrowth, and amenability. Dynamical systems and differential equations (Wilmington, NC, 2002). Discrete and Continuous Dynamical Systems, Series A. 2003, suppl., pp. 678-687.
  43. Richard Sharp. Critical exponents for groups of isometries. Geometriae Dedicata, vol. 125 (2007), pp. 63-74
  44. Laureates of the Moscow Mathematical Society Prize
  45. Fulbright Scholar Directory
  46. International Conference on GROUP THEORY: combinatorial, geometric, and dynamical aspects of infinite groups. Archived 2010-12-12 at the Wayback Machine
  47. Preface, International Journal of Algebra and Computation, vol. 15 (2005), no. 5-6, pp. v-vi
  48. Bartholdi, L., Ceccherini-Silberstein, T., Smirnova-Nagnibeda, T., Zuk, A.Infinite Groups: Geometric, Combinatorial and Dynamical Aspects.
  49. [https://dof.tamu.edu/dof/media/PITO-DOF/AFS%20DAA/AFS-DAA-University-Level-All-Winners_1.pdf Recipients of The Association of Former Students Distinguished Achievement Awards University Level]
  50. List of Fellows of the American Mathematical Society, retrieved 2013-01-19.
  51. AMS 2015 Leroy P. Steele Prize
  52. Laureates of Bogolyubov Prize of Ukrainian Academy of Science
  53. Laureate of Humboldt Research Award
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