Mladen Bestvina

Mladen Bestvina (born 1959[1]) 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.

Mladen Bestvina in 1986

Biographical info

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).[2] He received a B. Sc. in 1982 from the University of Zagreb.[3] He obtained a PhD in Mathematics in 1984 at the University of Tennessee under the direction of John Walsh.[4] He was a visiting scholar at the Institute for Advanced Study in 1987-88 and again in 1990–91.[5] Bestvina had been a faculty member at UCLA, and joined the faculty in the Department of Mathematics at the University of Utah in 1993.[6] He was appointed a Distinguished Professor at the University of Utah in 2008.[6] Bestvina received the Alfred P. Sloan Fellowship in 1988–89[7][8] and a Presidential Young Investigator Award in 1988–91.[9]

Bestvina gave an invited address at the International Congress of Mathematicians in Beijing in 2002.[10] He also gave a Unni Namboodiri Lecture in Geometry and Topology at the University of Chicago.[11]

Bestvina served as an Editorial Board member for the Transactions of the American Mathematical Society[12] and as an associate editor of the Annals of Mathematics.[13] Currently he is an Editorial Board member for Duke Mathematical Journal,[14]Geometric and Functional Analysis,[15] the Journal of Topology and Analysis,[16] Groups, Geometry and Dynamics,[17] Michigan Mathematical Journal,[18] Rocky Mountain Journal of Mathematics,[19] and Glasnik Matematicki.[20]

In 2012 he became a fellow of the American Mathematical Society.[21]

Mathematical contributions

A 1988 monograph of Bestvina[22] 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".'[23]

In a 1992 paper Bestvina and Feighn obtained a Combination Theorem for word-hyperbolic groups.[24] 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.[25][26][27][28]).

Bestvina and Feighn also gave the first published treatment of Rips' theory of stable group actions on R-trees (the Rips machine)[29] In particular their paper gives a proof of the Morgan–Shalen conjecture[30] 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).[31] In the same paper they introduced the notion of a relative train track and applied train track methods to solve[31] 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[32] 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[33] 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;[34] and others.

Bestvina, Feighn and Handel later proved that the group Out(Fn) satisfies the Tits alternative,[35][36] settling a long-standing open problem.

In a 1997 paper[37] 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.[38]

Selected publications
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See also

References
  1. "Mladen Bestvina". info.hazu.hr (in Croatian). Croatian Academy of Sciences and Arts. Retrieved 2013-03-29.
  2. "Mladen Bestvina". imo-official.org. International Mathematical Olympiad. Retrieved 2010-02-10.
  3. Research brochure: Mladen Bestvina, Department of Mathematics, University of Utah. Accessed February 8, 2010
  4. Mladen F. Bestvina, Mathematics Genealogy Project. Accessed February 8, 2010.
  5. Institute for Advanced Study: A Community of Scholars
  6. Mladen Bestvina: Distinguished Professor, Aftermath, vol. 8, no. 4, April 2008. Department of Mathematics, University of Utah.
  7. Sloan Fellows. Department of Mathematics, University of Utah. Accessed February 8, 2010
  8. Sloan Research Fellowships, Archived 2011-04-24 at the Wayback Machine Alfred P. Sloan Foundation. Accessed February 8, 2010
  9. Award Abstract #8857452. Mathematical Sciences: Presidential Young Investigator. National Science Foundation. Accessed February 8, 2010
  10. Invited Speakers for ICM2002. Notices of the American Mathematical Society, vol. 48, no. 11, December 2001; pp. 1343 1345
  11. Annual Lecture Series. Archived 2010-06-09 at the Wayback Machine Department of Mathematics, University of Chicago. Accessed February 9, 2010
  12. Officers and Committee Members, Notices of the American Mathematical Society, vol. 54, no. 9, October 2007, pp. 1178 1187
  13. Editorial Board, Archived 2009-05-19 at Archive.today Annals of Mathematics. Accessed February 8, 2010
  14. Duke Mathematical Journal
  15. Editorial Board, Geometric and Functional Analysis. Accessed February 8, 2010
  16. Editorial Board. Journal of Topology and Analysis. Accessed February 8, 2010
  17. Editorial Board, Groups, Geometry and Dynamics. Accessed February 8, 2010
  18. Editorial Board, Michigan Mathematical Journal. Accessed February 8, 2010
  19. Editorial Board, ROCKY MOUNTAIN JOURNAL OF MATHEMATICS. Accessed February 8, 2010
  20. Editorial Board, Glasnik Matematicki. Accessed February 8, 2010
  21. List of Fellows of the American Mathematical Society, retrieved 2012-11-10.
  22. Bestvina, Mladen, Characterizing k-dimensional universal Menger compacta. Memoirs of the American Mathematical Society, vol. 71 (1988), no. 380
  23. John J. Walsh, Review of: Bestvina, Mladen, Characterizing k-dimensional universal Menger compacta. Mathematical Reviews, MR0920964 (89g:54083), 1989
  24. M. Bestvina and M. Feighn, A combination theorem for negatively curved groups. Journal of Differential Geometry, Volume 35 (1992), pp. 85–101
  25. EMINA ALIBEGOVIC, A COMBINATION THEOREM FOR RELATIVELY HYPERBOLIC GROUPS. Bulletin of the London Mathematical Society vol. 37 (2005), pp. 459–466
  26. Francois Dahmani, Combination of convergence groups. Geometry and Topology, Volume 7 (2003), 933–963
  27. I. Kapovich, The combination theorem and quasiconvexity. International Journal of Algebra and Computation, Volume: 11 (2001), no. 2, pp. 185–216
  28. M. Mitra, Cannon–Thurston maps for trees of hyperbolic metric spaces. Journal of Differential Geometry, Volume 48 (1998), Number 1, 135–164
  29. M. Bestvina and M. Feighn. Stable actions of groups on real trees. Inventiones Mathematicae, vol. 121 (1995), no. 2, pp. 287 321
  30. Morgan, John W., Shalen, Peter B., Free actions of surface groups on R-trees. Topology, vol. 30 (1991), no. 2, pp. 143–154
  31. Mladen Bestvina, and Michael Handel, Train tracks and automorphisms of free groups. Annals of Mathematics (2), vol. 135 (1992), no. 1, pp. 1–51
  32. P. Brinkmann, Hyperbolic automorphisms of free groups. Geometric and Functional Analysis, vol. 10 (2000), no. 5, pp. 1071–1089
  33. Martin R. Bridson and Daniel Groves. The quadratic isoperimetric inequality for mapping tori of free-group automorphisms. Memoirs of the American Mathematical Society, to appear.
  34. 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
  35. Mladen Bestvina, Mark Feighn, and Michael Handel. The Tits alternative for Out(Fn). I. Dynamics of exponentially-growing automorphisms. Archived 2011-06-06 at the Wayback Machine Annals of Mathematics (2), vol. 151 (2000), no. 2, pp. 517–623
  36. 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
  37. Bestvina, Mladen and Brady, Noel, Morse theory and finiteness properties of groups. Inventiones Mathematicae, vol. 129 (1997), no. 3, pp. 445–470
  38. 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
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