arXiv subjects

Ryan Dickmann

Publications and source records attributed to Ryan Dickmann.

9 recordsLinked to original sources

arXiv2026

Large flats in large subgraphs of fine curve graphs

The fine curve graph of a surface is a graph whose vertices are essential simple closed curves and whose edges connect disjoint curves. Following a rich history of hyperbolicity of various graphs associated to surfaces, the fine curve graph was shown to be hyperbolic by Bowden-Hensel-Webb, while the curve graph, obtained from the fine curve graph by collapsing subgraphs corresponding to isotopy classes, was first proven to be hyperbolic by Masur-Minsky. We show that certain large subgraphs of fine curve graphs, including fibers over a vertex of the curve graph, are not hyperbolic. Indeed, such graphs contain flats of every finite dimension. We then compute bounds on distances in fibers over a vertex of the curve graph, which we call single-isotopy-class fine curve graphs.

math.GT↗

arXiv2026

Homotopy types of fine curve and fine arc complexes

The fine curve complex of a surface is a simplicial complex whose vertices are essential simple closed curves and whose $k$-simplices are collections of $k+1$ disjoint curves. We prove that the fine curve complex is homotopy equivalent to the curve complex. We also prove that the fine arc complex is contractible.

math.GT↗

arXiv2025

Geodesic-preserving bijections of the Thurston geometries

We completely classify the bijections of the Thurston geometries that preserve geodesics as sets. For Riemannian manifolds that satisfy a certain technical condition, we prove that a totally geodesic subset is a submanifold. We also classify the geodesic-preserving bijections of the Euclidean cylinder $\mathbb{S}^1 \times \mathbb{R}$ and the bijections of the hyperbolic plane $\mathbb{H}^2$ that preserve constant curvature curves.

math.GT↗

arXiv2024

Surfaces proper homotopy equivalent to graphs and their Dehn-Nielsen-Baer maps

Motivated by the recent work of Algom-Kfir and Bestinva introducing the mapping class group of an infinite graph via proper homotopy equivalences, we give a necessary and sufficient condition for a surface to be properly homotopy equivalent to a graph. We consider second-countable orientable surfaces that are possibly infinite-type and have noncompact boundary. For surfaces proper homotopy equivalent to graphs, we explore the basic properties of the induced map between the mapping class groups of the surface and the graph. We view this induced map as the basis of a Dehn-Nielsen-Baer analog in the setting of infinite-type surfaces.

math.GT↗

arXiv2023

Automatic continuity of pure mapping class groups

We completely classify the orientable infinite-type surfaces $S$ such that $\operatorname{PMap}(S)$, the pure mapping class group, has automatic continuity. This classification includes surfaces with noncompact boundary. In the case of surfaces with finitely many ends and no noncompact boundary components, we prove the mapping class group $\operatorname{Map}(S)$ does not have automatic continuity. We also completely classify the surfaces such that $\overline{\operatorname{PMap}_c(S)}$, the subgroup of the pure mapping class group composed of elements with representatives that can be approximated by compactly supported homeomorphisms, has automatic continuity. In some cases when $\overline{\operatorname{PMap}_c(S)}$ has automatic continuity, we show any homomorphism from $\overline{\operatorname{PMap}_c(S)}$ to a countable group is trivial.

math.GT↗

arXiv2022

Thurston's Theorem: Entropy in Dimension One

In his paper, Thurston shows that a positive real number $h$ is the topological entropy for an ergodic traintrack representative of an outer automorphism of a free group if and only if its expansion constant $\lambda = e^h$ is a weak Perron number. This is a powerful result, answering a question analogous to one regarding surfaces and stretch factors of pseudo-Anosov homeomorphisms. However, much of the machinery used to prove this seminal theorem on traintrack maps is contained in the part of Thurston's paper on the entropy of postcritically finite interval maps and the proof difficult to parse. In this expository paper, we modernize Thurston's approach, fill in gaps in the original paper, and distill Thurston's methods to give a cohesive proof of the traintrack theorem. Of particular note is the addition of a proof of ergodicity of the traintrack representatives, which was missing in Thurston's paper.

math.GT↗

arXiv2021

Mapping class groups of surfaces with noncompact boundary components

We show that the pure mapping class group is uniformly perfect for a certain class of infinite type surfaces with noncompact boundary components. We then combine this result with recent work in the remaining cases to give a complete classification of the perfect and uniformly perfect pure mapping class groups for infinite type surfaces. We also develop a method to cut a general surface into simpler surfaces and extend some mapping class group results to the general case.

math.GT↗

arXiv2020

Free products from spinning and rotating families

The far-reaching work of Dahmani-Guirardel-Osin and recent work of Clay-Mangahas-Margalit provide geometric approaches to the study of the normal closure of a subgroup (or a collection of subgroups)in an ambient group $G$. Their work gives conditions under which the normal closure in $G$ is a free product. In this paper we unify their results and simplify and significantly shorten the proof of the Dahmani-Guirardel-Osin theorem.

math.GT↗

arXiv2020

Big pure mapping class groups are never perfect

We show that the closure of the compactly supported mapping class group of an infinite type surface is not perfect and that its abelianization contains a direct summand isomorphic to an uncountable direct sum of rationals. We also extend this to the Torelli group and show that in the case of surfaces with infinite genus the abelianization of the Torelli group contains an indivisible copy of an uncountable free abelian group as well. Finally we give an application to the question of automatic continuity by exhibiting discontinuous homomorphisms to the rationals.

math.GT↗