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Mladen Bestvina

Publications and source records attributed to Mladen Bestvina.

At least 37 records · Page 2Linked to original sources
arXiv2020

Proper actions on finite products of quasi-trees

We say that a finitely generated group $G$ has property (QT) if it acts isometrically on a finite product of quasi-trees so that orbit maps are quasi-isometric embeddings. A quasi-tree is a connected graph with path metric quasi-isometric to a tree, and product spaces are equipped with the $\ell^1$-metric. As an application of the projection complex techniques, we prove that residually finite hyperbolic groups and mapping class groups have (QT).

math.GR↗
arXiv2018

A compactification of outer space which is an absolute retract

We define a new compactification of outer space $CV_N$ (the \emph{Pacman compactification}) which is an absolute retract, for which the boundary is a $Z$-set. The classical compactification $\overline{CV_N}$ made of very small $F_N$-actions on $\mathbb{R}$-trees, however, fails to be locally $4$-connected as soon as $N\ge 4$. The Pacman compactification is a blow-up of $\overline{CV_N}$, obtained by assigning an orientation to every arc with nontrivial stabilizer in the trees.

math.GT↗
arXiv2018

The Farrell-Jones Conjecture for mapping class groups

We prove the Farrell-Jones Conjecture for mapping class groups. The proof uses the Masur-Minsky theory of the large scale geometry of mapping class groups and the geometry of the thick part of Teichmueller space. The proof is presented in an axiomatic setup, extending the projection axioms of Bestvina-Bromberg-Fujiwara. More specifically, we prove that the action of the mapping class group on the Thurston compactification of Teichmueller space is finitely F-amenable for the family F consisting of virtual point stabilizers.

math.GT↗
arXiv2017

Acylindrical actions on projection complexes

We simplify the construction of projection complexes due to Bestvina-Bromberg-Fujiwara. To do so, we introduce a sharper version of the Behrstock inequality, and show that it can always be enforced. Furthermore, we use the new setup to prove acylindricity results for the action on the projection complexes. We also treat quasi-trees of metric spaces associated to projection complexes, and prove an acylindricity criterion in that context as well.

math.GR↗
arXiv2017

Bounded contractions for affine buildings

We consider affine buildings with refined chamber structure. For each vertex in the refined chamber structure we construct a contraction, based at the vertex, that is used to prove exactness of Schneider-Stuhler resolutions of arbitrary depth.

math.RT↗
arXiv2016

Handlebody subgroups in a mapping class group

Suppose subgroups $A,B < MCG(S)$ in the mapping class group of a closed orientable surface $S$ are given and let $\langle A, B \rangle$ be the subgroup they generate. We discuss a question by Minsky asking when $\langle A, B \rangle \simeq A*_{A \cap B} B$ for handlebody subgroups $A,B$.

math.GT↗
arXiv2015

Homological stability of $Aut(F_n)$ revisited

We give another proof of a theorem of Hatcher and Vogtmann stating that the sequence $Aut(F_n)$ satisfies integral homological stability. The paper is for the most part expository, and we also explain Quillen's method for proving homological stability.

math.GT↗
arXiv2015

Quasiflats in CAT(0) complexes

We show that if X is a piecewise Euclidean 2-complex with a cocompact isometry group, then every 2-quasiflat in X is at finite Hausdorff distance from a subset which is locally flat outside a compact set, and asymptotically conical.

math.GR↗
arXiv2015

Bounded cohomology with coefficients in uniformly convex Banach spaces

We show that for acylindrically hyperbolic groups $Γ$ (with no nontrivial finite normal subgroups) and arbitrary unitary representation $ρ$ of $Γ$ in a (nonzero) uniformly convex Banach space the vector space $H^2_b(Γ;ρ)$ is infinite dimensional. The result was known for the regular representations on $\ell^p(Γ)$ with $1<p<\infty$ by a different argument. But our result is new even for a non-abelian free group in this great generality for representations, and also the case for acylindrically hyperbolic groups follows as an application.

math.GR↗
arXiv2014

Constructing group actions on quasi-trees and applications to mapping class groups

A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general construction of many actions of groups on quasi-trees. The groups we can handle include non-elementary (relatively) hyperbolic groups, rank 1 CAT(0) groups, mapping class groups and Out(Fn). As an application, we show that mapping class groups act on finite products of δ-hyperbolic spaces so that orbit maps are quasi-isometric embeddings. We prove that mapping class groups have finite asymptotic dimension.

math.GR↗
arXiv2013

Stable commutator length on mapping class groups

Let $Γ$ be a finite index subgroup of the mapping class group $MCG(Σ)$ of a closed orientable surface $Σ$, possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element $g\inΓ$ has positive stable commutator length. In addition, we show that in these situations the stable commutator length, if nonzero, is uniformly bounded away from 0. The method works for certain subgroups of infinite index as well and we show $scl$ is uniformly positive on the nontrivial elements of the Torelli group. The proofs use our earlier construction in the paper "Constructing group actions on quasi-trees and applications to mapping class groups" of group actions on quasi-trees.

math.GR↗
arXiv2013

The boundary of the complex of free factors

We give a description of the boundary of a complex of free factors that is analogous to E. Klarreich's description of the boundary of a curve complex. The argument uses the geometry of folding paths developed by Bestvina and Feighn as well as structural results about very small trees developed by Coulbois, Hilion, Lustig, and Reynolds.

math.GR↗
arXiv2013

Subfactor projections

When two free factors A and B of a free group F_n are in "general position" we define the projection of B to the splitting complex (alternatively, the complex of free factors) of A. We show that the projections satisfy properties analogous to subsurface projections introduced by Masur and Minsky. We use the subfactor projections to construct an action of Out(F_n) on a finite product of hyperbolic spaces where every automorphism with exponential growth acts with positive translation length. We also prove a version of the Bounded geodesic image theorem. In the appendix, we give a sketch of the proof of the Handel-Mosher hyperbolicity theorem for the splitting complex using (liberal) folding paths.

math.GR↗