arXiv subjects

Spencer Dowdall

Publications and source records attributed to Spencer Dowdall.

22 records · Page 2Linked to original sources

arXiv2013

Statistical hyperbolicity in Teichmüller space

In this paper we explore the idea that Teichmüller space is hyperbolic "on average." Our approach focuses on studying the geometry of geodesics which spend a definite proportion of time in some thick part of Teichmüller space. We consider several different measures on Teichmüller space and find that this behavior for geodesics is indeed typical. With respect to each of these measures, we show that the average distance between points in a ball of radius r is asymptotic to 2r, which is as large as possible. Our techniques also lead to a statement quantifying the expected thinness of random triangles in Teichmüller space, showing that "most triangles are mostly thin."

math.GT↗

arXiv2013

Pseudo-Anosov subgroups of fibered 3-manifold groups

Let X be a hyperbolic surface and H the fundamental group of a hyperbolic 3-manifold that fibers over the circle with fiber X. Using the Birman exact sequence, H embeds in the mapping class group Mod(Y) of the surface Y obtained by removing a point from X. We prove that a subgroup G in H is convex cocompact in Mod(Y) if and only if G is finitely generated and purely pseudo-Anosov. We also prove a generalization of this theorem with H replaced by an arbitrary Gromov hyperbolic extension of the fundamental group of X, and an additional hypothesis of quasi-convexity of G in H. Along the way, we obtain a generalization of a theorem of Scott and Swarup on the geometric finiteness of subgroups of fibered 3-manifold groups.

math.GT↗

arXiv2012

Spheres in the curve complex

In this paper we study the geometry of metric spheres in the curve complex of a surface, with the goal of determining the "average" distance between points on a given sphere. Averaging is not technically possible because metric spheres in the curve complex are countably infinite and do not support any invariant probability measures. To make sense of the idea of averaging, we instead develop definitions of null and generic subsets in a way that is compatible with the topological structure of the curve complex. With respect to this notion of genericity, we show that pairs of points on a sphere of radius R almost always have distance exactly 2R apart, which is as large as possible.

math.GT↗

arXiv2011

Dilatation versus self-intersection number for point-pushing pseudo-Anosov homeomorphisms

A filling curve $γ$ on a based surface $S$ determines a pseudo-Anosov homeomorphism $P(γ)$ of $S$ via the process of "point-pushing along $γ$." We consider the relationship between the self-intersection number $i(γ)$ of $γ$ and the dilatation of $P(γ)$; our main result is that the dilatation is bounded between $(i(γ)+1)^{1/5}$ and $9^{i(γ)}$. We also bound the least dilatation of any pseudo-Anosov in the point-pushing subgroup of a closed surface and prove that this number tends to infinity with genus. Lastly, we investigate the minimal entropy of any pseudo-Anosov homeomorphism obtained by pushing along a curve with self-intersection number $k$ and show that, for a closed surface, this number grows like $\log(k)$.

math.GT↗