arXiv subjects

Matthew Gentry Durham

Publications and source records attributed to Matthew Gentry Durham.

13 recordsLinked to original sources

arXiv2025

Bicombing the mapping class group and Teichm\"uller space via stable cubical intervals

In this mostly expository article, we provide a new account of our proof with Minsky and Sisto that mapping class groups and Teichm\"uller spaces admit bicombings. More generally, we explain how the hierarchical hull of a pair of points in any colorable hierarchically hyperbolic space is quasi-isometric to a finite CAT(0) cube complex of bounded dimension, with the added property that perturbing the pair of points results in a uniformly bounded change to the cubical structure. Our approach is simplified and new in many aspects.

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arXiv2025

Atypical generic directions in Teichm\"uller space

Motivated by geometrically capturing generic directions in Teichm\"uller space -- that is, tracking rays for random walks of the mapping class group -- we use work of Chaika--Masur--Wolf and Durham--Zalloum to construct the first examples of a sublinearly-Morse Teichm\"uller geodesic rays with minimal non-uniquely ergodic vertical foliations.

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arXiv2025

Asymptotically CAT(0) metrics, Z-structures, and the Farrell-Jones Conjecture

We show that colorable hierarchically hyperbolic groups (HHGs) admit asymptotically CAT(0) metrics, that is, roughly, metrics where the CAT(0) inequality holds up to sublinear error in the size of the triangle. We use the asymptotically CAT(0) metrics to construct contractible simplicial complexes and compactifications that provide $\mathcal{Z}$-structures in the sense of Bestvina and Dranishnikov. It was previously unknown that mapping class groups are asymptotically CAT(0) and admit $\mathcal{Z}$-structures. As an application, we prove that many HHGs satisfy the Farrell--Jones Conjecture, including extra large-type Artin groups. To construct asymptotically CAT(0) metrics, we show that hulls of finitely many points in a colorable HHGs can be approximated by CAT(0) cube complexes in a way that adding a point to the finite set corresponds, up to finitely many hyperplanes deletions, to a convex embedding.

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arXiv2023

Cubulating Infinity in Hierarchically Hyperbolic Spaces

We prove that the hierarchical hull of any finite set of interior points, hierarchy rays, and boundary points in a hierarchically hyperbolic space (HHS) is quasi-median quasi-isometric to a CAT(0) cube complex of bounded dimension. Our construction extends and refines a theorem of Behrstock-Hagen-Sisto about modeling hulls of interior points and our previous work with Zalloum on modeling finite sets of rays via limits of these finite models. We further prove that the quasi-median quasi-isometry between the hull of a finite set of rays or boundary points and its cubical model extends to an isomorphism between their respective hierarchical and simplicial boundaries. In this sense, we prove that the hierarchical boundary of any proper HHS is locally modeled by the simplicial boundaries of CAT(0) cube complexes. This is a purely geometric statement, allowing one to important various topologies from the cubical setting. Our proof of the cubical model theorem is new, even for the interior points case. In particular, we provide a concrete description of the cubical model as a cubical subcomplex of a product of simplicial trees into which the hierarchical data is directly encoded. Moreover, the above boundary isomorphism is new for all non-cubical HHSes, including mapping class groups and Teichmüller spaces of finite-type surfaces. As an application of our techniques, we show that in most HHSes, including all hierarchically hyperbolic groups, the distance between any pair of points in the top-level hyperbolic space is coarsely the length of a maximal 0-separated chain of hyperplanes separating them in an appropriate cubical model. For mapping class groups, this says that these cubical models cubically encode distance in the curve graph of the surface.

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arXiv2022

The geometry of genericity in mapping class groups and Teichmüller spaces via CAT(0) cube complexes

Random walks on spaces with hyperbolic properties tend to sublinearly track geodesic rays which point in certain hyperbolic-like directions. Qing-Rafi-Tiozzo recently introduced the sublinearly Morse boundary and proved that this boundary is a quasi-isometry invariant which captures a notion of generic direction in a broad context. In this article, we develop the geometric foundations of sublinear Morseness in the mapping class group and Teichmüller space. We prove that their sublinearly Morse boundaries are visibility spaces and admit continuous equivariant injections into the boundary of the curve graph. Moreover, we completely characterize sublinear Morseness in terms of the hierarchical structures of these spaces. Our techniques include developing tools for modeling the hulls of median rays in hierarchically hyperbolic spaces via CAT(0) cube complexes. Part of this analysis involves establishing direct connections between the geometry of the curve graph and the combinatorics of hyperplanes in the approximating cube complexes.

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arXiv2020

Largest acylindrical actions and stability in hierarchically hyperbolic groups

We consider two manifestations of non-positive curvature: acylindrical actions on hyperbolic spaces and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for studying many important families of groups, including mapping class groups, right-angled Coxeter and Artin groups, most 3-manifold groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces, so it is natural to search for a "best" one. The set of all cobounded acylindrical actions on hyperbolic spaces admits a natural poset structure; in this paper we prove that all hierarchically hyperbolic groups admit a unique action which is the largest in this poset. The action we construct is also universal in the sense that every element which acts loxodromically in some acylindrical action on a hyperbolic space does so in this one. Special cases of this result are themselves new and interesting. For instance, this is the first proof that right-angled Coxeter groups admit universal acylindrical actions. The notion of quasigeodesic stability of subgroups provides a natural analogue of quasiconvexity outside the context of hyperbolic groups. We provide a complete classification of stable subgroups of hierarchically hyperbolic groups, generalizing and extending results that are known for mapping class groups and right-angled Artin groups. We also provide a characterization of contracting quasigeodesics; interestingly, in this generality the proof is much simpler than in the special cases where it was already known. In the appendix, it is verified that any space satisfying the a priori weaker property of being an "almost hierarchically hyperbolic space" is actually a hierarchically hyperbolic space. The results of the appendix are used to streamline the proofs in the main text.

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arXiv2017

Pulling back stability with applications to Out($F_n$) and relatively hyperbolic groups

We prove that stability -- a strong quasiconvexity property -- pulls back under proper actions on proper metric spaces. This result has several applications, including that convex cocompact subgroups of both mapping class groups and outer automorphism groups of free groups are stable. We also characterize stability in relatively hyperbolic groups whose parabolic subgroups have linear divergence.

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arXiv2016

Graphs of curves on infinite-type surfaces with mapping class group actions

We study when the mapping class group of an infinite-type surface $S$ admits an action with unbounded orbits on a connected graph whose vertices are simple closed curves on $S$. We introduce a topological invariant for infinite-type surfaces that determines in many cases whether there is such an action. This allows us to conclude that, as non-locally compact topological groups, many big mapping class groups have nontrivial coarse geometry in the sense of Rosendal.

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arXiv2016

Boundary convex cocompactness and stability of subgroups of finitely generated groups

A Kleinian group $Γ< \mathrm{Isom}(\mathbb H^3)$ is called convex cocompact if any orbit of $Γ$ in $\mathbb H^3$ is quasiconvex or, equivalently, $Γ$ acts cocompactly on the convex hull of its limit set in $\partial \mathbb H^3$. Subgroup stability is a strong quasiconvexity condition in finitely generated groups which is intrinsic to the geometry of the ambient group and generalizes the classical quasiconvexity condition above. Importantly, it coincides with quasiconvexity in hyperbolic groups and convex cocompactness in mapping class groups. Using the Morse boundary, we develop an equivalent characterization of subgroup stability which generalizes the above boundary characterization from Kleinian groups.

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arXiv2015

The augmented marking complex of a surface

We build an augmentation of the Masur-Minsky marking complex by Groves-Manning combinatorial horoballs to obtain a graph we call the augmented marking complex, $\mathcal{AM}(S)$. Adapting work of Masur-Minsky, we prove that $\mathcal{AM}(S)$ is quasiisometric to Teichmüller space with the Teichmüller metric. A similar construction was independently discovered by Eskin-Masur-Rafi. We also completely integrate the Masur-Minsky hierarchy machinery to $\mathcal{AM}(S)$ to build flexible families of uniform quasigeodesics in Teichmüller space. As an application, we give a new proof of Rafi's distance formula for the Teichmüller metric.

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arXiv2014

The asymptotic geometry of the Teichmüller metric: Dimension and rank

We analyze the asymptotic cones of Teichmüller space with the Teichmüller metric, $(\mathcal{T}(S),d_T)$. We give a new proof of a theorem of Eskin-Masur-Rafi which bounds the dimension of quasiisometrically embedded flats in $(\mathcal{T}(S),d_T)$. Our approach is an application of the ideas of Behrstock and Behrstock-Minsky to the quasiisometry model we previously built for $(\mathcal{T}(S),d_T)$.

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arXiv2014

Elliptic actions on Teichmuller space

Let $S$ be an oriented surface of finite type, $\mathcal{MCG}(S)$ its mapping class group, and $\mathcal{T}(S)$ its Teichmüller space with the Teichmüller metric. Let $H \leq \mathcal{MCG}(S)$ be a finite subgroup and consider the subset of $\mathcal{T}(S)$ fixed by $H$, $\mathrm{Fix}(H) \subset \mathcal{T}(S)$. For any $R>0$, we prove that the set of points whose $H$-orbits have diameter bounded by $R$, $\mathrm{Fix}_R^T(H)$, lives in a bounded neighborhood of $\mathrm{Fix}(H)$. As an application, we show that the orbit of any point $X \in \mathcal{T}(S)$ under the action of a finite order mapping class has a fixed coarse barycenter. By contrast, we show that $\mathrm{Fix}^T_R(H)$ need not be quasiconvex with an explicit family of examples.

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arXiv2014

Convex cocompactness and stability in mapping class groups

We introduce a strong notion of quasiconvexity in finitely generated groups, which we call stability. Stability agrees with quasiconvexity in hyperbolic groups and is preserved under quasi-isometry for finitely generated groups. We show that the stable subgroups of mapping class groups are precisely the convex cocompact subgroups. This generalizes a well-known result of Behrstock and is related to questions asked by Farb-Mosher and Farb.

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