Mladen Bestvina (born 1959) is a Croatian-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.
Life and career Mladen Bestvina is a three-time medalist at the International Mathematical Olympiad (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.Research brochure: Mladen Bestvina, Department of Mathematics, University of Utah. Accessed February 8, 2010 He obtained a PhD in Mathematics in 1984 at the University of Tennessee under the direction of John Walsh.Mladen F. Bestvina, Mathematics Genealogy Project. Accessed February 8, 2010. He was a visiting scholar at the Institute for Advanced Study in 1987-88 and again in 1990–91. Bestvina had been a faculty member at UCLA, and joined the faculty in the Department of Mathematics at the University of Utah in 1993.Mladen Bestvina: Distinguished Professor, Aftermath, vol. 8, no. 4, April 2008. Department of Mathematics, University of Utah. He was appointed a Distinguished Professor at the University of Utah in 2008. Bestvina received the Alfred P. Sloan Fellowship in 1988–89Sloan Fellows. Department of Mathematics, University of Utah. Accessed February 8, 2010Sloan Research Fellowships, Alfred P. Sloan Foundation. Accessed February 8, 2010 and a Presidential Young Investigator Award in 1988–91.Award Abstract #8857452. Mathematical Sciences: Presidential Young Investigator. National Science Foundation. Accessed February 8, 2010
Bestvina gave an invited address at the International Congress of Mathematicians in Beijing in 2002,Invited Speakers for ICM2002. Notices of the American Mathematical Society, vol. 48, no. 11, December 2001; pp. 1343 1345, was topology section member in Madrid in 2006 Program Committee and Sectional Panels 2006 Madrid, Spain, Sectional Panels Topology and gave a plenary lecture at virtual ICM 2022.ICM Plenary and Invited Speakers Mladen Bestvina He also gave a Unni Namboodiri Lecture in Geometry and Topology at the University of Chicago.Annual Lecture Series. Department of Mathematics, University of Chicago. Accessed February 9, 2010
Bestvina served as an Editorial Board member for the Transactions of the American Mathematical SocietyOfficers and Committee Members, Notices of the American Mathematical Society, vol. 54, no. 9, October 2007, pp. 1178 1187 and as an associate editor of the Annals of Mathematics.Editorial Board, Annals of Mathematics. Accessed February 8, 2010 Currently he is an editorial board member for Duke Mathematical Journal, Geometric and Functional Analysis,Editorial Board, Geometric and Functional Analysis. Accessed February 8, 2010 Geometry and Topology,Editorial Board Geometry and Topology the Journal of Topology and Analysis,Editorial Board. Journal of Topology and Analysis. Accessed February 8, 2010 Groups, Geometry and Dynamics,Editorial Board, Groups, Geometry and Dynamics. Accessed February 8, 2010 Michigan Mathematical Journal,Editorial Board, Michigan Mathematical Journal. Accessed February 8, 2010 Rocky Mountain Journal of Mathematics,Editorial Board, Rocky Mountain Journal of Mathematics. Accessed February 8, 2010 and Glasnik Matematicki.Editorial Board, Glasnik Matematicki. Accessed February 8, 2010
In 2012 he became a fellow of the American Mathematical Society.List of Fellows of the American Mathematical Society, retrieved 2012-11-10. Since 2012, he has been a correspondent member of the HAZU (Croatian Academy of Science and Art).
Mathematical contributions A 1988 monograph of BestvinaBestvina, Mladen, Characterizing k-dimensional universal Menger compacta. Memoirs of the American Mathematical Society, vol. 71 (1988), no. 380 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 Tennessee, 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".'John J. Walsh, Review of: Bestvina, Mladen, Characterizing k-dimensional universal Menger compacta. Mathematical Reviews, MR0920964 (89g:54083), 1989
In a 1992 paper Bestvina and Feighn obtained a Combination Theorem for word-hyperbolic groups.M. Bestvina and M. Feighn, A combination theorem for negatively curved groups. Journal of Differential Geometry, Volume 35 (1992), pp. 85–101 The theorem provides a set of sufficient conditions for amalgamated free products and HNN extensions 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.Emina ALibegovic, A Combination Theorem for Relatively Hyperbolic Groups. Bulletin of the London Mathematical Society vol. 37 (2005), pp. 459–466Francois Dahmani, Combination of convergence groups. Geometry and Topology, Volume 7 (2003), 933–963I. Kapovich, The combination theorem and quasiconvexity. International Journal of Algebra and Computation, Volume: 11 (2001), no. 2, pp. 185–216M. Mitra, Cannon–Thurston maps for trees of hyperbolic metric spaces. Journal of Differential Geometry, Volume 48 (1998), Number 1, 135–164).
Bestvina and Feighn also gave the first published treatment of Rips' theory of stable group actions on R-trees (the Rips machine)M. Bestvina and M. Feighn. Stable actions of groups on real trees. Inventiones Mathematicae, vol. 121 (1995), no. 2, pp. 287 321 In particular their paper gives a proof of the Morgan–Shalen conjectureMorgan, John W., Shalen, Peter B., Free actions of surface groups on R-trees. Topology, vol. 30 (1991), no. 2, pp. 143–154 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 groups and free abelian groups.
A 1992 paper of Bestvina and Handel introduced the notion of a train track map for representing elements of Out(Fn).Mladen Bestvina, and Michael Handel, Train tracks and automorphisms of free groups. Annals of Mathematics (2), vol. 135 (1992), no. 1, pp. 1–51 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 BrinkmannP. Brinkmann, Hyperbolic automorphisms of free groups. Geometric and Functional Analysis, vol. 10 (2000), no. 5, pp. 1071–1089 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 GrovesMartin R. Bridson and Daniel Groves. The quadratic isoperimetric inequality for mapping tori of free-group automorphisms. Memoirs of the American Mathematical Society, to appear. 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;O. Bogopolski, A. Martino, O. Maslakova, E. Ventura, The conjugacy problem is solvable in free-by-cyclic groups. Bulletin of the London Mathematical Society, vol. 38 (2006), no. 5, pp. 787–794 and others.
Bestvina, Feighn and Handel later proved that the group Out(Fn) satisfies the Tits alternative,Mladen Bestvina, Mark Feighn, and Michael Handel. The Tits alternative for Out(Fn). I. Dynamics of exponentially-growing automorphisms. Annals of Mathematics (2), vol. 151 (2000), no. 2, pp. 517–623Mladen Bestvina, Mark Feighn, and Michael Handel. The Tits alternative for Out(Fn). II. A Kolchin type theorem. Annals of Mathematics (2), vol. 161 (2005), no. 1, pp. 1–59 settling a long-standing open problem.
In a 1997 paperBestvina, Mladen and Brady, Noel, Morse theory and finiteness properties of groups. Inventiones Mathematicae, vol. 129 (1997), no. 3, pp. 445–470 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.Brady, Noel, Branched coverings of cubical complexes and subgroups of hyperbolic groups. Journal of the London Mathematical Society (2), vol. 60 (1999), no. 2, pp. 461–480
Selected publications Bestvina, Mladen, Characterizing k-dimensional universal Menger compacta. Memoirs of the American Mathematical Society, vol. 71 (1988), no. 380 Bestvina, Mladen; Feighn, Mark, Bounding the complexity of simplicial group actions on trees. Inventiones Mathematicae, vol. 103 (1991), no. 3, pp. 449–469 Bestvina, Mladen; Mess, Geoffrey, The boundary of negatively curved groups. Journal of the American Mathematical Society, vol. 4 (1991), no. 3, pp. 469–481 Mladen Bestvina, and Michael Handel, Train tracks and automorphisms of free groups. Annals of Mathematics (2), vol. 135 (1992), no. 1, pp. 1–51 M. Bestvina and M. Feighn, A combination theorem for negatively curved groups. Journal of Differential Geometry, Volume 35 (1992), pp. 85–101 M. Bestvina and M. Feighn. Stable actions of groups on real trees. Inventiones Mathematicae, vol. 121 (1995), no. 2, pp. 287 321 Bestvina, Mladen and Brady, Noel, Morse theory and finiteness properties of groups. Inventiones Mathematicae, 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 Mathematics (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 Mathematics (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 Society, vol. 23 (2010), no. 1, pp. 61–105
See also Real tree Artin group Out(Fn) Train track map Pseudo-Anosov map Word-hyperbolic group Mapping class group Whitehead conjecture
References
External links Mladen Bestvina, personal webpage, Department of Mathematics, University of Utah