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]
Mathematical contributions
edit
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]).
A 1992 paper of Bestvina and Handel introduced the notion of a train track map for representing elements of Out(Fn).[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 groupFn 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[34] 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[35] 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;[36] and others.
Bestvina, Feighn and Handel later proved that the group Out(Fn) 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]
Selected publications
edit
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
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
^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.
^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
^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 MachineAnnals 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 and Brady, Noel, Morse theory and finiteness properties of groups. Inventiones Mathematicae, vol. 129 (1997), no. 3, pp. 445–470
^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
External links
edit
Mladen Bestvina, personal webpage, Department of Mathematics, University of Utah