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Jason Behrstock

Publications and source records attributed to Jason Behrstock.

27 records · Page 2Linked to original sources
arXiv2011

Divergence, thick groups, and short conjugators

In this paper we explore relationships between divergence and thick groups, and with the same techniques we estimate lengths of shortest conjugators. We produce examples, for every positive integer n, of CAT(0) groups which are thick of order n and with polynomial divergence of order n+1, both these phenomena are new. With respect to thickness, these examples show the non-triviality at each level of the thickness hierarchy defined by Behrstock-Drutu-Mosher. With respect to divergence our examples resolve questions of Gromov and Gersten (the divergence questions were also recently and independently answered by Macura. We also provide general tools for obtaining both lower and upper bounds on the divergence of geodesics and spaces, and we give the definitive lower bound for Morse geodesics in the CAT(0) spaces, generalizing earlier results of Kapovich-Leeb and Bestvina-Fujiwara. In the final section, we turn to the question of bounding the length of the shortest conjugators in several interesting classes of groups. We obtain linear and quadratic bounds on such lengths for classes of groups including 3-manifold groups and mapping class groups (the latter gives new proofs of corresponding results of Masur-Minsky in the pseudo-Anosov case and Tao in the reducible case).

math.GT↗
arXiv2010

Homomorphisms into mapping class groups. An addendum

This is an addendum to arXiv: 0810.5376. We show, using our methods and an auxiliary result of Bestvina-Bromberg-Fujiwara, that a finitely generated group with infinitely many pairwise non-conjugate homomorphisms to a mapping class group virtually acts non-trivially on an $\R$-tree, and, if it is finitely presented, it virtually acts non-trivially on a simplicial tree

math.GR↗
arXiv2010

Geometry and rigidity of mapping class groups

We study the large scale geometry of mapping class groups MCG(S), using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG(S) (outside a few sporadic cases) is a bounded distance away from a left-multiplication, and as a consequence obtain quasi-isometric rigidity for MCG(S), namely that groups quasi-isometric to MCG(S) are virtually equal to it. (The latter theorem was proved by Hamenstadt using different methods). As part of our approach we obtain several other structural results: a description of the tree-graded structure on the asymptotic cone of MCG(S); a characterization of the image of the curve-complex projection map from MCG(S) to the product of the curve complexes of essential subsurfaces of S; and a construction of Sigma-hulls in MCG(S), an analogue of convex hulls.

math.GT↗
arXiv2010

Quasi-isometric classification of non-geometric 3-manifold groups

We describe the quasi-isometric classification of fundamental groups of irreducible non-geometric 3-manifolds which do not have "too many" arithmetic hyperbolic geometric components, thus completing the quasi-isometric classification of 3--manifold groups in all but a few exceptional cases.

math.GT↗
arXiv2010

Divergence and quasimorphisms of right-angled Artin groups

We give a group theoretic characterization of geodesics with superlinear divergence in the Cayley graph of a right-angled Artin group A(G) with connected defining graph G. We use this to determine when two points in an asymptotic cone of A(G) are separated by a cut-point. As an application, we show that if G does not decompose as the join of two subgraphs, then A(G) has an infinite-dimensional space of non-trivial quasimorphisms. By the work of Burger and Monod, this leads to a superrigidity theorem for homomorphisms from lattices into right-angled Artin groups.

math.GR↗
arXiv2009

Growth of intersection numbers for free group automorphisms

For a fully irreducible automorphism ϕof the free group F_k we compute the asymptotics of the intersection number n \mapsto i(T,T'ϕ^n) for trees T,T' in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and T'ϕ^n for n large.

math.GR↗
arXiv2008

Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with nonrelatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficient for a metric space to be nonhyperbolic relative to any nontrivial collection of subsets. Thick finitely generated groups include: mapping class groups of most surfaces; outer automorphism groups of most free groups; certain Artin groups; and others. Nonuniform lattices in higher rank semisimple Lie groups are thick and hence nonrelatively hyperbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples of nonrelatively hyperbolic groups having cut points in any asymptotic cone, resolving several questions of Drutu and Sapir about the structure of relatively hyperbolic groups. Outside of group theory, Teichmuller spaces for surfaces of sufficiently large complexity are thick with respect to the Weil-Peterson metric, in contrast with Brock--Farb's hyperbolicity result in low complexity.

math.GT↗
arXiv2005

Curve complexes and finite index subgroups of mapping class groups

Let Mod(S) be the extended mapping class group of a surface S. For S the twice-punctured torus, we show that there exists an isomorphism of finite index subgroups of Mod(S) which is not the restriction of an inner automorphism. For S a torus with at least three punctures, we show that every injection of a finite index subgroup of Mod(S) into Mod(S) is the restriction of an inner automorphism; this completes a program begun by Irmak. For all of the above surfaces, we establish the co-Hopf property for finite index subgroups of Mod(S).

math.GT↗