470 research outputs found
Invariance of coarse median spaces under relative hyperbolicity
We show that, for finitely generated groups, the property of admitting a coarse median structure is preserved under relative hyperbolicity
Large-scale rank and rigidity of the Weil-Petersson metric
We study the large-scale geometry of WeilâPetersson space, that is, TeichmuĚller space equipped with theWeilâPetersson metric. We show that this admits a natural coarse median structure of a specific rank. Given that this is equal to the maximal dimension of a quasi-isometrically embedded euclidean space,we recover a result of Eskin,Masur and Rafi which gives the coarse rank of the space. We go on to show that, apart from finitely many cases, the WeilâPetersson spaces are quasi-isometrically distinct, and quasi-isometrically rigid. In particular, any quasi-isometry between such spaces is a bounded distance from an isometry. By a theorem of Brock,WeilâPetersson space is equivariantly quasi-isometric to the pants graph, so our results apply equally well to that space
Peripheral splittings of groups
We define the notion of a "peripheral splitting" of a group. This
is essentially a representation of the group as the fundamental group of a bipartite
graph of groups, where all the vertex groups of one colour are held
fixed - the "peripheral subgroups". We develop the theory of such splittings
and prove an accessibility result. The theory mainly applies to relatively hyperbolic
groups with connected boundary, where the peripheral subgroups are
precisely the maximal parabolic subgroups. We show that if such a group admits
a non-trivial peripheral splitting, then its boundary has a global cut point.
Moreover, the non-peripheral vertex groups of such a splitting are themselves
relatively hyperbolic. These results, together with results from elsewhere, show
that under modest constraints on the peripheral subgroups, the boundary of
a relatively hyperbolic group is locally connected if it is connected. In retrospect,
one further deduces that the set of global cut points in such a boundary
has a simplicial treelike structure
Rigidity of the strongly separating curve graph
We define the strongly separating curve graph to be the full subgraph of the curve graph of a compact orientable surface, where the vertex set consists of all separating curves which do not bound a three-holed sphere. We show that, for all but finitely many surfaces, any automorphism of the strongly separating curve graph is induced by an element of the mapping class group
Median and injective metric spaces
We describe a construction which associates to any median metric space a pseudometric satisfying the binary intersection property for closed balls. Under certain conditions, this implies that the resulting space is, in fact, an injective metric space, bilipschitz equivalent to the original metric. In the course of doing this, we derive a few other facts about median metrics, and the geometry of CAT(0) cube complexes. One motivation for the study of such metrics is that they arise as asymptotic cones of certain naturally occurring spaces
Some properties of median metric spaces
We describe a number of properties of a median metric space. In particular, we show that a complete connected median metric space of finite rank admits a canonical bi-lipschitz equivalent CAT(0) metric. Metric spaces of this sort arise, up to bi-lipschitz equivalence, as asymptotic cones of certain classes of finitely generated groups, and the existence of such a structure has various consequences for the large scale geometry of the group
Large-scale rigidity properties of the mapping class groups
We study the coarse geometry of the mapping class group of a
compact orientable surface. We show that, apart from a few low-complexity cases, any quasi-isometric embedding of a mapping class group into itself agrees up to bounded distance with a left multiplication. In particular, such a map is a quasi-isometry. This is a strengthening of the result of Hamenst¨adt and of Behrstock, Kleiner, Minsky and Mosher that the mapping class groups are quasi-isometrically rigid. In the course of proving this, we also develop the
general theory of coarse median spaces and median metric spaces with a view to applications to Teichm¨uller space, and related spaces
Quasi-isometric maps between direct products of hyperbolic spaces
We give conditions under which a quasi-isometric map between direct products of hyperbolic spaces splits as a direct product up to bounded distance and permutation of factors. This is a variation on a result due to Kapovich, Kleiner and Leeb
- âŚ