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Harold Mark Sultan

Publications and source records attributed to Harold Mark Sultan.

5 recordsLinked to original sources
arXiv2012

Explicit equivalences between CAT(0) hyperbolic type geodesics

We prove an explicit equivalence between various hyperbolic type properties for quasi-geodesics in CAT(0) spaces. Specifically, we prove that for X a CAT(0) space and $γ$ a quasi-geodesic, the following four statements are equivalent and moreover the quantifiers in the equivalences are explicit: (i) $γ$ is S-Slim, (ii) $γ$ is M-Morse, (iii) $γ$ is (b,c)-contracting, and (iv) $γ$ is C-strongly contracting. In particular, this explicit equivalence proves that for $f$ a (K,L)-quasi-isometry between CAT(0) spaces, and $γ$ a C-strongly contracting (K',L')-quasi-geodesic, then $f(γ)$ is a C'(C,K,L,K',L')-strongly contracting quasi-geodesic. This result is necessary for a key technical point with regard to Charney's contracting boundary for CAT(0) spaces.

math.GT↗
arXiv2012

The Asymptotic Cone of Teichmüller Space: Thickness and Divergence

We study the Asymptotic Cone of Teichmüller space equipped with the Weil-Petersson metric. In particular, we provide a characterization of the canonical finest pieces in the tree-graded structure of the asymptotic cone of Teichmüller space along the same lines as a similar characterization for right angled Artin groups by Behrstock-Charney and for mapping class groups by Behrstock-Kleiner-Minksy-Mosher. As a corollary of the characterization, we complete the thickness classification of Teichmüller spaces for all surfaces of finite type, thereby answering questions of Behrstock-Drutu, Behrstock-Drutu-Mosher, and Brock-Masur. In particular, we prove that Teichmüller space of the genus two surface with one boundary component (or puncture) can be uniquely characterized in the following two senses: it is thick of order two, and it has superquadratic yet at most cubic divergence. In addition, we characterize strongly contracting quasi-geodesics in Teichmüller space, generalizing results of Brock-Masur-Minsky. As a tool, we develop a complex of separating multicurves, which may be of independent interest.

math.GT↗
arXiv2011

Hyperbolic quasi-geodesics in CAT(0) spaces

We prove that in CAT(0) spaces a quasi-geodesic is Morse if and only if it is contracting. Specifically, in our main theorem we prove that for $γ$ a quasi-geodesic in a CAT(0) space X, the following four statements are equivalent: (i) $γ$ is Morse, (ii) $γ$ is (b,c)--contracting, (iii), $γ$ is strongly contracting, and (iv) in every asymptotic cone $X_ω,$ any two distinct points in the ultralimit $γ_ω$ are separated by a cutpoint. As a corollary, we provide a converse to the usual Morse stability lemma in the CAT(0) setting. In addition, as a warm up we include an alternative proof of the fact that in CAT(0) spaces Morse quasi-geodesics have at least quadratic divergence, originally proven by Behrstock-Drutu.

math.GT↗
arXiv2011

Separating Pants Decompositions in the Pants Complex

We study the topological types of pants decompositions of a surface by associating to any pants decomposition $P,$ in a natural way its pants decomposition graph, $Γ(P).$ This perspective provides a convenient way to analyze the maximum distance in the pants complex of any pants decomposition to a pants decomposition containing a non-trivial separating curve for all surfaces of finite type. In the main theorem we provide an asymptotically sharp approximation of this non-trivial distance in terms of the topology of the surface. In particular, for closed surfaces of genus $g$ we show the maximum distance in the pants complex of any pants decomposition to a pants decomposition containing a separating curve grows asymptotically like the function $\log(g).$

math.GT↗