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
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]
^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