Rostislav Grigorchuk

Rostislav Ivanovich Grigorchuk ( _ru. Ростислав Иванович Григорчук)(b. February 23, 1953) is a Ukrainian mathematician working in the area of group theory. He holds the rank of Distinguished Professor in the Mathematics Department of the Texas A&M University. Grigorchuk is particularly well-known for having constructed, in a 1984 paper, 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 groupPierre de la Harpe. "Topics in geometric group theory." Chicago Lectures in Mathematics. University of Chicago Press, Chicago. ISBN: 0-226-31719-6 ] [Laurent Bartholdi. "The growth of Grigorchuk's torsion group." International Mathematics Research Notices, 1998, no. 20, pp. 1049-1054 ] [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 ] [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] [Roman Muchnik, and Igor Pak. "Percolation on Grigorchuk groups." Communications in Algebra, vol. 29 (2001), no. 2, pp. 661-671. ] and it is one of the important objects studied in geometric group theory, particularly in the study of branch groups, automata groups and iterated monodromy groups.

Biographical data

Grigorchuk was born on February 23, 1953 in Ternopol oblast, Ukraine.He received his undergraduate degree in 1975 from the Moscow State University.He obtained a PhD (Candidate of Science) in Mathematics in 1978, also from the Moscow State University, where his thesis advisor was A. M. Stepin. Grigorchuk received a habilitation (Doctor of Science) degree in Mathematics in 1985 at the Steklov Institute of Mathematics in Moscow. During 1980s and 1990s Rostislav Grigorchuk held positions at the Moscow State University of Transportation and, subsequently at the Steklov Institute of Mathematics and the Moscow State University. In 2002 Grigorchuk joined the faculty of the Texas A&M University as a Professor of Mathematics and he was promoted to the rank of Distinguished Professor in 2008. [ [http://www.math.tamu.edu/news_events/newsletter/2008pers.html 2008 TAMU Math Faculty News] ]

Rostislav Grigorchuk gave an invited address at the 1990 International Congress of Mathematicians in Kyoto. [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 ]

In June 2003 an international group theory conference in honor of Grigorchuk's 50th birthday was held in Gaeta, Italy. [ [http://www.math.kth.se/gaeta/ International Conference on GROUP THEORY: combinatorial, geometric, and dynamical aspects of infinite groups.] ] Special anniversary issues of the "International Journal of Algebra and Computation" and of the journal "Algebra and Discrete Mathematics" were dedicated to Grigorchuk's 50th birthday. [ [http://www.worldscinet.com/cgi-bin/details.cgi?id=jsname:ijac&type=all Preface] , International Journal of Algebra and Computation, vol. 15 (2005), no. 5-6, pp. v-vi ] [http://adm.lnpu.edu.ua/issues/adm_n4_2003_frame.htm Editorial Statement in honor of Grigorchuk's 50th birthday.] Algebra and Discrete Mathematics, (2003), no. 4]

Grigorchuk is the Editor-in-Chief on the journal "Groups, Geometry and Dynamics" [ [http://www.math.tamu.edu/journals/ggd/ "Groups, Geometry and Dynamics"] ] , published by the European Mathematical Society, and a member of editorial boards for the journals "International Journal of Algebra and Computation" [ [http://www.worldscinet.com/ijac/mkt/editorial.shtml Editorial Board, International Journal of Algebra and Computation] ] , "Journal of Modern Dynamics" [ [http://www.math.psu.edu/jmd/EditorialBoard.html Editorial Board, Journal of Moden Dynamics] ] , "Geometriae Dedicata" [ [http://www.springer.com/math/geometry/journal/10711?detailsPage=editorialBoard Editorial Board, Geometriae Dedicata] ] , "Algebra and Discrete Mathematics" [ [http://adm.lnpu.edu.ua/editorial/index_frame.htm Editorial Board, Algebra and Discrete Mathematics] ] and "Matematychni Studii".

Mathematical contributions

Grigorchuk is most well-known for having constucted 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 [R. I. Grigorchuk. "On Burnside's problem on periodic groups." (Russian) Funktsionalyi Analiz i ego Prilozheniya, vol. 14 (1980), no. 1, pp. 53-54] and proved that it has intermediate growth in a 1984 articleR. 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] . 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 [ Mahlon M. Day. "Amenable semigroups."
Illinois Journal of Mathematics, vol. 1 (1957), pp. 509-544.] . Also Grigorchuk's group is "just infinite": that is, it is infinite but every proper quotient of this group is finite.

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 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. [Volodymyr Nekrashevych. "Self-similar groups." Mathematical Surveys and Monographs, 117. American Mathematical Society, Providence, RI, 2005. ISBN: 0-8218-3831-8]

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. [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. ] [R. I. Grigorchuk, P. Linnell, T. Schick, and A. Zuk. "On a question of Atiyah." Comptes Rendus de l'Académie des Sciences. Série I. Mathématique. vol. 331 (2000), no. 9, pp. 663-668.]

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 [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 ] what is commonly known (see for example [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] [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.] [Richard Sharp. "Critical exponents for groups of isometries." Geometriae Dedicata, vol. 125 (2007), pp. 63-74] ) as Grigorchuk's co-growth criterion of amenability for finitely generated groups.

ee also

*Geometric group theory
*Growth of groups
*Iterated monodromy group
*Amenable groups

References

External links

* [http://www.math.tamu.edu/~grigorch/ Web-page of Rostislav Grigorchuk at Texas A&M University]