Rostislav Grigorchuk
Encyclopedia
Rostislav Ivanovich Grigorchuk is a Soviet and Ukrainian
mathematician working in the area of group theory
. 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, 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
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 group
s.
, Ukraine
.
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 A. M. Stepin. Grigorchuk received a habilitation (Doctor of Science) degree in Mathematics in 1985 at the Steklov Institute of Mathematics
in Moscow
. 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
. 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.
Rostislav Grigorchuk gave an invited address at the 1990 International Congress of Mathematicians
in Kyoto
an AMS Invited Address at the March 2004 meeting of the American Mathematical Society
in Athens, Ohio and a plenary talk at the 2004 Winter Meeting of the Canadian Mathematical Society
.
In June 2003 an international group theory
conference in honor of Grigorchuk's 50th birthday was held in Gaeta, Italy. 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.
Grigorchuk is the Editor-in-Chief of the journal "Groups, Geometry and Dynamics", published by the European Mathematical Society
, and a member of the editorial boards of the journals "International Journal of Algebra and Computation", "Journal of Modern Dynamics", "Geometriae Dedicata", "Algebra and Discrete Mathematics" and "Matematychni Studii".
(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 and proved that it has intermediate growth in a 1984 article. 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. 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 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, fractal
s, Hecke algebra
s, bounded cohomology, functional analysis
, and others. In particular, many of these self-similar groups arise as iterated monodromy group
s 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 set
s.
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.
Grigorchuk is also known for his contributions to the general theory of random walks on groups and the theory of amenable group
s, particularly for obtaining in 1980 what is commonly known (see for example ) as Grigorchuk's co-growth criterion of amenability for finitely generated groups.
Ukraine
Ukraine is a country in Eastern Europe. It has an area of 603,628 km², making it the second largest contiguous country on the European continent, after Russia...
mathematician working in the area of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
. He holds the rank of Distinguished Professor
Professors in the United States
In the U.S., "Professors" commonly occupy any of several positions in academia, typically the ranks of Assistant Professor, Associate Professor or Full Professor....
in the Mathematics Department of Texas A&M University
Texas A&M University
Texas A&M University is a coeducational public research university located in College Station, Texas . It is the flagship institution of the Texas A&M University System. The sixth-largest university in the United States, A&M's enrollment for Fall 2011 was over 50,000 for the first time in school...
. Grigorchuk is particularly well known for having constructed, in a 1984 paper, the first example of a finitely generated group of intermediate growth
Growth rate (group theory)
In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length...
, thus answering an important problem posed by John Milnor
John Milnor
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...
in 1968. This group is now known as the Grigorchuk group
Grigorchuk group
In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate growth...
and it is one of the important objects studied in geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
, particularly in the study of branch groups, automata groups and iterated monodromy group
Iterated monodromy group
In geometric group theory and dynamical systems the iterated monodromy group of a covering map is a group describing the monodromy action of the fundamental group on all iterations of the covering...
s.
Biographical data
Grigorchuk was born on February 23, 1953 in Ternopol oblastTernopil Oblast
Ternopil Oblast is an oblast' of Ukraine. Its administrative center is Ternopil, through which flows the Seret River, a tributary of the Dnister.-Geography:...
, Ukraine
Ukraine
Ukraine is a country in Eastern Europe. It has an area of 603,628 km², making it the second largest contiguous country on the European continent, after Russia...
.
He received his undergraduate degree in 1975 from Moscow State University
Moscow State University
Lomonosov Moscow State University , previously known as Lomonosov University or MSU , is the largest university in Russia. Founded in 1755, it also claims to be one of the oldest university in Russia and to have the tallest educational building in the world. Its current rector is Viktor Sadovnichiy...
.
He obtained a PhD (Candidate of Science) in Mathematics in 1978, also from Moscow State University
Moscow State University
Lomonosov Moscow State University , previously known as Lomonosov University or MSU , is the largest university in Russia. Founded in 1755, it also claims to be one of the oldest university in Russia and to have the tallest educational building in the world. Its current rector is Viktor Sadovnichiy...
, 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
Steklov Institute of Mathematics
Steklov Institute of Mathematics or Steklov Mathematical Institute is a research institute based in Moscow, specialized in mathematics, and a part of the Russian Academy of Sciences. It was established April 24, 1934 by the decision of the General Assembly of the Academy of Sciences of the USSR in...
in Moscow
Moscow
Moscow is the capital, the most populous city, and the most populous federal subject of Russia. The city is a major political, economic, cultural, scientific, religious, financial, educational, and transportation centre of Russia and the continent...
. During the 1980s and 1990s, Rostislav Grigorchuk held positions at the Moscow State University of Transportation
Moscow State University of Railway Engineering
Moscow State University of Railway Engineering is the leading and largest higher education institution in the field of railway transport in Russia with more than 17,000...
, and subsequently at the Steklov Institute of Mathematics
Steklov Institute of Mathematics
Steklov Institute of Mathematics or Steklov Mathematical Institute is a research institute based in Moscow, specialized in mathematics, and a part of the Russian Academy of Sciences. It was established April 24, 1934 by the decision of the General Assembly of the Academy of Sciences of the USSR in...
and Moscow State University
Moscow State University
Lomonosov Moscow State University , previously known as Lomonosov University or MSU , is the largest university in Russia. Founded in 1755, it also claims to be one of the oldest university in Russia and to have the tallest educational building in the world. Its current rector is Viktor Sadovnichiy...
. In 2002 Grigorchuk joined the faculty of Texas A&M University
Texas A&M University
Texas A&M University is a coeducational public research university located in College Station, Texas . It is the flagship institution of the Texas A&M University System. The sixth-largest university in the United States, A&M's enrollment for Fall 2011 was over 50,000 for the first time in school...
as a Professor of Mathematics, and he was promoted to the rank of Distinguished Professor in 2008.
Rostislav Grigorchuk gave an invited address at the 1990 International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
in Kyoto
Kyoto
is a city in the central part of the island of Honshū, Japan. It has a population close to 1.5 million. Formerly the imperial capital of Japan, it is now the capital of Kyoto Prefecture, as well as a major part of the Osaka-Kobe-Kyoto metropolitan area.-History:...
an AMS Invited Address at the March 2004 meeting of the American Mathematical Society
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
in Athens, Ohio and a plenary talk at the 2004 Winter Meeting of the Canadian Mathematical Society
Canadian Mathematical Society
The Canadian Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and education in Canada.It was originally conceived in June 1945 as the Canadian Mathematical Congress...
.
In June 2003 an international group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
conference in honor of Grigorchuk's 50th birthday was held in Gaeta, Italy. 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.
Grigorchuk is the Editor-in-Chief of the journal "Groups, Geometry and Dynamics", published by the European Mathematical Society
European Mathematical Society
The European Mathematical Society is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians...
, and a member of the editorial boards of the journals "International Journal of Algebra and Computation", "Journal of Modern Dynamics", "Geometriae Dedicata", "Algebra and Discrete Mathematics" and "Matematychni Studii".
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 groupGrigorchuk group
In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate growth...
(sometimes it is also called the first Grigorchuk group since Grigorchuk constructed several other groups that are also commonly studied). This group has growth
Growth rate (group theory)
In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length...
that is faster than polynomial but slower than exponential. Grigorchuk constructed this group in a 1980 paper and proved that it has intermediate growth in a 1984 article. This result answered a long-standing open problem posed by John Milnor
John Milnor
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...
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
Residually finite group
In the mathematical field of group theory, a group G is residually finite or finitely approximable if for every nontrivial element g in G there is a homomorphism h from G to a finite group, such thath \neq 1.\,...
2-group
P-group
In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...
(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
Amenable group
In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements...
but not elementary amenable
Elementary amenable group
In mathematics, a group is called elementary amenable if it can be built up from finite groups and abelian groups by a sequence of simple operations that result in amenable groups when applied to amenable groups...
, thus providing an answer to another long-standing problem, posed by Mahlon Day in 1957. Also Grigorchuk's group is "just infinite": that is, it is infinite but every proper quotient
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
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 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
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
, ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
, random walks, fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
s, Hecke algebra
Hecke algebra
In mathematics, the Iwahori–Hecke algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a one-parameter deformation of the group algebra of a Coxeter group....
s, bounded cohomology, functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, and others. In particular, many of these self-similar groups arise as iterated monodromy group
Iterated monodromy group
In geometric group theory and dynamical systems the iterated monodromy group of a covering map is a group describing the monodromy action of the fundamental group on all iterations of the covering...
s 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 set
Julia set
In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...
s.
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
Michael Atiyah
Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...
about L2-betti numbers of closed manifolds.
Grigorchuk is also known for his contributions to the general theory of random walks on groups and the theory of amenable group
Amenable group
In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements...
s, particularly for obtaining in 1980 what is commonly known (see for example ) as Grigorchuk's co-growth criterion of amenability for finitely generated groups.
See also
- Geometric group theoryGeometric group theoryGeometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
- Growth of groupsGrowth rate (group theory)In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length...
- Iterated monodromy groupIterated monodromy groupIn geometric group theory and dynamical systems the iterated monodromy group of a covering map is a group describing the monodromy action of the fundamental group on all iterations of the covering...
- Amenable groupAmenable groupIn mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements...
s - Grigorchuk groupGrigorchuk groupIn the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate growth...