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Mladen Bestvina

Croatian-American mathematician

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 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 MathematiciansinBeijing in 2002,^[10] and gave a plenary lecture at virtual ICM 2022.^[11] He also gave a Unni Namboodiri Lecture in Geometry and Topology at the University of Chicago.^[12]

Bestvina served as an Editorial Board member for the Transactions of the American Mathematical Society^[13] and as an associate editor of the Annals of Mathematics.^[14] Currently he is an editorial board member for Duke Mathematical Journal,^[15] Geometric and Functional Analysis,^[16] Geometry and Topology,^[17] the Journal of Topology and Analysis,^[18] Groups, Geometry and Dynamics,^[19] Michigan Mathematical Journal,^[20] Rocky Mountain Journal of Mathematics,^[21] and Glasnik Matematicki.^[22]

In 2012 he became a fellow of the American Mathematical Society.^[23] Since 2012, he has been a correspondent member of the HAZU (Croatian Academy of Science and Art).^[1]

A 1988 monograph of Bestvina^[24] 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".'^[25]

In a 1992 paper Bestvina and Feighn obtained a Combination Theorem for word-hyperbolic groups.^[26] 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.^{[27][28][29][30]}).

Bestvina and Feighn also gave the first published treatment of Rips' theory of stable group actions on R-trees (the Rips machine)^[31] In particular their paper gives a proof of the Morgan–Shalen conjecture^[32] 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(F_n).^[33] In the same paper they introduced the notion of a relative train track and applied train track methods to solve^[33] the Scott conjecture, which says that for every automorphism α of a finitely generated free group F_n 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(F_n). Examples of applications of train tracks include: a theorem of Brinkmann^[34] proving that for an automorphism αofF_n the mapping torus group of αisword-hyperbolic if and only if α has no periodic conjugacy classes; a theorem of Bridson and Groves^[35] that for every automorphism αofF_n 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;^[36] and others.

Bestvina, Feighn and Handel later proved that the group Out(F_n) satisfies the Tits alternative,^[37][38] settling a long-standing open problem.

In a 1997 paper^[39] 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.^[40]

• 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(F_n). 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(F_n). 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

• Real tree
• Artin group
• Out(F_n)
• Train track map
• Pseudo-Anosov map
• Word-hyperbolic group
• Mapping class group
• Whitehead conjecture

• Mladen Bestvina, personal webpage, Department of Mathematics, University of Utah