Mladen Bestvina
Encyclopedia
Mladen Bestvina is a Croatia
n American mathematician
working in the area of geometric group theory
. He is a Distinguished Professor in the Department of Mathematics at the University of Utah
.
(two silver medals in 1976 and 1978 and a bronze medal in 1977). He received a B. Sc. in 1982 from the University of Zagreb
. He obtained a PhD in Mathematics in 1984 at the University of Tennessee
under the direction of John Walsh. Bestvina had been a faculty member at UCLA, and joined the faculty in the Department of Mathematics at the University of Utah
in 1993. He was appointed a Distinguished Professor at the University of Utah
in 2008.
Bestvina received the Alfred P. Sloan Fellowship
in 1988–89 and a Presidential Young Investigator Award
in 1988–91.
Bestvina gave an Invited Address at the International Congress of Mathematicians
in Beijing
in 2002.
He also gave a Unni Namboodiri Lecture in Geometry and Topology at the University of Chicago
.
Bestvina served as an Editorial Board member for the Transactions of the American Mathematical Society
.
He is currently an associate editor of the Annals of Mathematics
and an Editorial Board member for Geometric and Functional Analysis, the Journal of Topology and Analysis, Groups, Geometry and Dynamics, Michigan Mathematical Journal, Rocky Mountain Journal of Mathematics, and Glasnik Matematicki.
, represents a monumental step forward, having moved the status of the topological structure of higher-dimensional Menger compacta from one of ``close to total ignorance" to one of ``complete understanding".
In a 1992 paper Bestvina and Feighn obtained a Combination Theorem for word-hyperbolic groups. The theorem provides a set of sufficient conditions for amalgamated free products
and HNN extension
s of word-hyperbolic groups to again be word-hyperbolic. The Bestvina–Feighn Combination Theorem became a standard tool in geometric group theory
and has had many applications and generalizations (e.g. ).
Bestvina and Feighn also gave the first published treatment of Rips'
theory of stable group actions on R-trees
(the Rips machine
) In particular their paper gives a proof of the Morgan–Shalen conjecture that a finitely generated group G admits a free isometric action
on an R-tree
if and only if G is a free product
of surface groups, free group
s and free abelian group
s.
A 1992 paper of Bestvina and Handel introduced the notion of a train track map
for representing elements of Out(Fn)
. In the same paper they introduced the notion of a relative train track and applied train track methods to solve the Scott conjecture which says that for every automorphism α of a finitely generated free group
Fn the fixed subgroup of α is free of rank at most n. Since then train tracks became a standard tool in the study of algebraic, geometric and dynamical properties of automorphisms of free groups and of subgroups of Out(Fn). Examples of applications of train tracks include: a theorem of Brinkmann proving that for an automorphism α of Fn the mapping torus group of α is word-hyperbolic if and only if α has no periodic conjugacy classes; a theorem of Bridson and Groves that for every automorphism α of Fn the mapping torus group of α satisfies a quadratic isoperimetric inequality
; a proof of algorithmic solvability of the conjugacy problem
for free-by-cyclic groups; and others.
Bestvina, Feighn and Handel later proved that the group Out(Fn) satisfies the Tits alternative
, settling a long-standing open problem.
In a 1997 paper Bestvina and Brady developed a version of discrete Morse theory for cubical complexes and applied it to study homological finiteness properties of subgroups of right-angled Artin groups. In particular, they constructed an example of a group which provides a counter-example to either the Whitehead asphericity conjecture
or to the Eilenberg−Ganea conjecture, thus showing that at least one of these conjectures must be false. Brady subsequently used their Morse theory technique to construct the first example of a finitely presented subgroup of a word-hyperbolic group that is not itself word-hyperbolic.
Croatia
Croatia , officially the Republic of Croatia , is a unitary democratic parliamentary republic in Europe at the crossroads of the Mitteleuropa, the Balkans, and the Mediterranean. Its capital and largest city is Zagreb. The country is divided into 20 counties and the city of Zagreb. Croatia covers ...
n American mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
working in the area of 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...
. He is a Distinguished Professor in the Department of Mathematics at the University of Utah
University of Utah
The University of Utah, also known as the U or the U of U, is a public, coeducational research university in Salt Lake City, Utah, United States. The university was established in 1850 as the University of Deseret by the General Assembly of the provisional State of Deseret, making it Utah's oldest...
.
Biographical info
Mladen Bestvina is a three-time medalist at the International Mathematical OlympiadInternational Mathematical Olympiad
The International Mathematical Olympiad is an annual six-problem, 42-point mathematical olympiad for pre-collegiate students and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, except in 1980...
(two silver medals in 1976 and 1978 and a bronze medal in 1977). He received a B. Sc. in 1982 from the University of Zagreb
University of Zagreb
The University of Zagreb is the biggest Croatian university and the oldest continuously operating university in the area covering Central Europe south of Vienna and all of Southeastern Europe...
. He obtained a PhD in Mathematics in 1984 at the University of Tennessee
University of Tennessee
The University of Tennessee is a public land-grant university headquartered at Knoxville, Tennessee, United States...
under the direction of John Walsh. Bestvina had been a faculty member at UCLA, and joined the faculty in the Department of Mathematics at the University of Utah
University of Utah
The University of Utah, also known as the U or the U of U, is a public, coeducational research university in Salt Lake City, Utah, United States. The university was established in 1850 as the University of Deseret by the General Assembly of the provisional State of Deseret, making it Utah's oldest...
in 1993. He was appointed a Distinguished Professor at the University of Utah
University of Utah
The University of Utah, also known as the U or the U of U, is a public, coeducational research university in Salt Lake City, Utah, United States. The university was established in 1850 as the University of Deseret by the General Assembly of the provisional State of Deseret, making it Utah's oldest...
in 2008.
Bestvina received the Alfred P. Sloan Fellowship
Sloan Fellowship
The Sloan Research Fellowships are awarded annually by the Alfred P. Sloan Foundation since 1955 to "provide support and recognition to early-career scientists and scholars". This is distinct from the Sloan Fellows in business....
in 1988–89 and a Presidential Young Investigator Award
PECASE
The Presidential Early Career Award for Scientists and Engineers is the highest honor bestowed by the United States government on outstanding scientists and engineers in the early stages of their independent research careers. The White House, following recommendations from participating agencies,...
in 1988–91.
Bestvina gave an Invited Address at the 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 Beijing
Beijing
Beijing , also known as Peking , is the capital of the People's Republic of China and one of the most populous cities in the world, with a population of 19,612,368 as of 2010. The city is the country's political, cultural, and educational center, and home to the headquarters for most of China's...
in 2002.
He also gave a Unni Namboodiri Lecture in Geometry and Topology at the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
.
Bestvina served as an Editorial Board member for the Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society is a monthly mathematics journal published by the American Mathematical Society. It started in 1900...
.
He is currently an associate editor of the Annals of Mathematics
Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
and an Editorial Board member for Geometric and Functional Analysis, the Journal of Topology and Analysis, Groups, Geometry and Dynamics, Michigan Mathematical Journal, Rocky Mountain Journal of Mathematics, and Glasnik Matematicki.
Mathematical contributions
A 1988 monograph of Bestvina gave an abstract topological characterization of universal Menger compacta in all dimensions; previously only the cases of dimension 0 and 1 were well understood. John Walsh wrote in a review of Bestvina's monograph:This work, which formed the author's Ph.D. thesis at the University of TennesseeUniversity of Tennessee
The University of Tennessee is a public land-grant university headquartered at Knoxville, Tennessee, United States...
, represents a monumental step forward, having moved the status of the topological structure of higher-dimensional Menger compacta from one of ``close to total ignorance" to one of ``complete understanding".
In a 1992 paper Bestvina and Feighn obtained a Combination Theorem for word-hyperbolic groups. The theorem provides a set of sufficient conditions for amalgamated free products
Free product
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...
and HNN extension
HNN extension
In mathematics, the HNN extension is a basic construction of combinatorial group theory.Introduced in a 1949 paper Embedding Theorems for Groups by Graham Higman, B. H...
s of word-hyperbolic groups to again be word-hyperbolic. The Bestvina–Feighn Combination Theorem became a standard tool 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...
and has had many applications and generalizations (e.g. ).
Bestvina and Feighn also gave the first published treatment of Rips'
Eliyahu Rips
Eliyahu Rips, also Ilya Rips is a Latvian-born Israeli mathematician known for his research in geometric group theory. He became known to the general public following his coauthoring a paper on the Torah Code....
theory of stable group actions on R-trees
Real tree
A real tree, or an \mathbb R-tree, is a metric space such thatfor any x, y in M there is a unique arc from x to y and this arc is a geodesic segment. Here by an arc from x to y we mean the image in M of a topological embedding f from an interval [a,b] to M such that f=x and f=y...
(the Rips machine
Rips machine
In geometric group theory, the Rips machine is a method of studying the action of groups on R-trees. It was introduced in unpublished work of Eliyahu Rips in about 1991....
) In particular their paper gives a proof of the Morgan–Shalen conjecture that a finitely generated group G admits a free isometric action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
on an R-tree
Real tree
A real tree, or an \mathbb R-tree, is a metric space such thatfor any x, y in M there is a unique arc from x to y and this arc is a geodesic segment. Here by an arc from x to y we mean the image in M of a topological embedding f from an interval [a,b] to M such that f=x and f=y...
if and only if G is a free product
Free product
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...
of surface groups, free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
s and free abelian group
Free abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
s.
A 1992 paper of Bestvina and Handel introduced the notion of a train track map
Train track map
In the mathematical subject of geometric group theory a train track map is a continuous map f from a finite connected graph to itself which is a homotopy equivalence and which has particularly nice cancellation properties with respect to iterations...
for representing elements of Out(Fn)
Out(Fn)
In mathematics, Out is the outer automorphism group of a free group on n generators. These groups play an important role in geometric group theory.-Outer space:...
. In the same paper they introduced the notion of a relative train track and applied train track methods to solve the Scott conjecture which says that for every automorphism α of a finitely generated free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
Fn the fixed subgroup of α is free of rank at most n. Since then train tracks became a standard tool in the study of algebraic, geometric and dynamical properties of automorphisms of free groups and of subgroups of Out(Fn). Examples of applications of train tracks include: a theorem of Brinkmann proving that for an automorphism α of Fn the mapping torus group of α is word-hyperbolic if and only if α has no periodic conjugacy classes; a theorem of Bridson and Groves that for every automorphism α of Fn the mapping torus group of α satisfies a quadratic isoperimetric inequality
Dehn function
In the mathematical subject of geometric group theory, a Dehn function, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the area of a relation in that group in terms of the length of that relation...
; a proof of algorithmic solvability of the conjugacy problem
Conjugacy problem
In abstract algebra, the conjugacy problem for a group G with a given presentation is the decision problem of determining, given two words x and y in G, whether or not they represent conjugate elements of G...
for free-by-cyclic groups; and others.
Bestvina, Feighn and Handel later proved that the group Out(Fn) satisfies the Tits alternative
Tits alternative
In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups. It states that every such group is either virtually solvable In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about...
, settling a long-standing open problem.
In a 1997 paper Bestvina and Brady developed a version of discrete Morse theory for cubical complexes and applied it to study homological finiteness properties of subgroups of right-angled Artin groups. In particular, they constructed an example of a group which provides a counter-example to either the Whitehead asphericity conjecture
Whitehead conjecture
The Whitehead conjecture is a claim in algebraic topology. It was formulated by J. H. C. Whitehead in 1941. It states that every connected subcomplex of a two-dimensional aspherical CW complex is aspherical....
or to the Eilenberg−Ganea conjecture, thus showing that at least one of these conjectures must be false. Brady subsequently used their Morse theory technique to construct the first example of a finitely presented subgroup of a word-hyperbolic group that is not itself word-hyperbolic.
Selected publications
- Bestvina, Mladen, Characterizing k-dimensional universal Menger compacta. Memoirs of the American Mathematical SocietyAmerican Mathematical SocietyThe 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...
, vol. 71 (1988), no. 380 - Bestvina, Mladen; Feighn, Mark, Bounding the complexity of simplicial group actions on trees. Inventiones MathematicaeInventiones MathematicaeInventiones Mathematicae, often just referred to as Inventiones, is a mathematical journal published monthly by Springer Verlag. It was founded in 1966.-Editors:...
, vol. 103 (1991), no. 3, pp. 449–469 - Bestvina, Mladen; Mess, Geoffrey, The boundary of negatively curved groups. Journal of the American Mathematical SocietyJournal of the American Mathematical SocietyThe Journal of the American Mathematical Society , is a quarterly, peer reviewed, mathematical journal published by the American Mathematical Society. The editors are Weinan E, Sergey Fomin, Gregory Lawler, John W. Morgan, Karl Rubin, and Terence Tao...
, vol. 4 (1991), no. 3, pp. 469–481 - Mladen Bestvina, and Michael Handel, Train tracks and automorphisms of free groups. Annals of MathematicsAnnals of MathematicsThe Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
(2), vol. 135 (1992), no. 1, pp. 1–51 - M. Bestvina and M. Feighn, A combination theorem for negatively curved groups. Journal of Differential GeometryJournal of Differential GeometryThe Journal of Differential Geometry is a peer-reviewed mathematical journal published by the Lehigh University. The journal publishes nine issues per year and is distributed by International Press. Surveys in Differential Geometry] It also publishes an annual supplement in book form called...
, Volume 35 (1992), pp. 85–101 - M. Bestvina and M. Feighn. Stable actions of groups on real trees. Inventiones MathematicaeInventiones MathematicaeInventiones Mathematicae, often just referred to as Inventiones, is a mathematical journal published monthly by Springer Verlag. It was founded in 1966.-Editors:...
, vol. 121 (1995), no. 2, pp. 287 321 - Bestvina, Mladen and Brady, Noel, Morse theory and finiteness properties of groups. Inventiones MathematicaeInventiones MathematicaeInventiones Mathematicae, often just referred to as Inventiones, is a mathematical journal published monthly by Springer Verlag. It was founded in 1966.-Editors:...
, vol. 129 (1997), no. 3, pp. 445–470 - Mladen Bestvina, Mark Feighn, and Michael Handel. The Tits alternative for Out(Fn). I. Dynamics of exponentially-growing automorphisms. Annals of MathematicsAnnals of MathematicsThe Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
(2), vol. 151 (2000), no. 2, pp. 517–623 - Mladen Bestvina, Mark Feighn, and Michael Handel. The Tits alternative for Out(Fn). II. A Kolchin type theorem. Annals of MathematicsAnnals of MathematicsThe Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
(2), vol. 161 (2005), no. 1, pp. 1–59 - Bestvina, Mladen; Bux, Kai-Uwe; Margalit, Dan, The dimension of the Torelli group. Journal of the American Mathematical SocietyJournal of the American Mathematical SocietyThe Journal of the American Mathematical Society , is a quarterly, peer reviewed, mathematical journal published by the American Mathematical Society. The editors are Weinan E, Sergey Fomin, Gregory Lawler, John W. Morgan, Karl Rubin, and Terence Tao...
, vol. 23 (2010), no. 1, pp. 61–105
See also
- Real treeReal treeA real tree, or an \mathbb R-tree, is a metric space such thatfor any x, y in M there is a unique arc from x to y and this arc is a geodesic segment. Here by an arc from x to y we mean the image in M of a topological embedding f from an interval [a,b] to M such that f=x and f=y...
- Artin group
- Out(Fn)Out(Fn)In mathematics, Out is the outer automorphism group of a free group on n generators. These groups play an important role in geometric group theory.-Outer space:...
- Train track mapTrain track mapIn the mathematical subject of geometric group theory a train track map is a continuous map f from a finite connected graph to itself which is a homotopy equivalence and which has particularly nice cancellation properties with respect to iterations...
- Pseudo-Anosov mapPseudo-Anosov mapIn mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface. It is a generalization of a linear Anosov diffeomorphism of the torus...
- Word-hyperbolic group
- Mapping class groupMapping class groupIn mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...
- Whitehead conjectureWhitehead conjectureThe Whitehead conjecture is a claim in algebraic topology. It was formulated by J. H. C. Whitehead in 1941. It states that every connected subcomplex of a two-dimensional aspherical CW complex is aspherical....
External links
- Mladen Bestvina, personal webpage, Department of Mathematics, University of UtahUniversity of UtahThe University of Utah, also known as the U or the U of U, is a public, coeducational research university in Salt Lake City, Utah, United States. The university was established in 1850 as the University of Deseret by the General Assembly of the provisional State of Deseret, making it Utah's oldest...