A 1988 monograph of Bestvina 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".' In a 1992 paper Bestvina and Feighn obtained a Combination Theorem for word-hyperbolic groups. 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. Bestvina and Feighn also gave the first published treatment of Rips' theory of stable group actions on R-trees In particular their paper gives a proof of the Morgan–Shalen conjecture that a finitely generated groupG admits a free isometric action on an R-tree if and only ifG 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. 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 generatedfree 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. Examples of applications of train tracks include: a theorem of Brinkmann 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 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; and others. Bestvina, Feighn and Handel later proved that the group Out satisfies the Tits alternative, settling a long-standing open problem. In a 1997 paper 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.
