arXiv subjects

Michael C. Kopreski

Publications and source records attributed to Michael C. Kopreski.

5 recordsLinked to original sources

arXiv2025

Geometric models and asymptotic dimension for infinite-type surface mapping class groups

Let $S$ be an infinite-type surface and let $G \leq \operatorname{Map}(S)$ be a locally bounded Polish subgroup. We construct a metric graph $M$ of simple arcs and curves on $S$ preserved by the action of $G$ and for which the vertex orbit map $G \to V(M)$ is a coarse equivalence; if $G$ is boundedly generated, then $M$ is a Cayley--Abels--Rosendal graph for $G$ and the orbit map is a quasi-isometry. In particular, if $S$ contains a non-displaceable subsurface and $G \geq \operatorname{PMap}_c(S)$ is boundedly generated or $G \in \{\overline{\operatorname{PMap}_c(S)}, \operatorname{PMap}(S), \operatorname{Map}(S) \}$ and is locally bounded, then $\operatorname{asdim} M = \operatorname{asdim} G = \infty$. This result completes the classification of the asymptotic dimension of stable boundedly generated infinite-type surface mapping class groups begun by Grant--Rafi--Verberne.

math.GT↗

arXiv2025

The asymptotic dimension of the grand arc graph is infinite

Let $Σ$ be a compact, orientable surface of genus $g$, and let $Γ$ be a relation on $π_0(\partial Σ)$ such that the prescribed arc graph $\mathcal{A}(Σ,Γ)$ is Gromov-hyperbolic and non-trivial. We show that $\operatorname{asdim} \mathcal{A}(Σ,Γ) \geq -χ(Σ) - 1$, from which we prove that the asymptotic dimension of the grand arc graph is infinite. More generally, an arc and curve model on $Σ$ is a graph of simple arc and curves on $Σ$, on which $\operatorname{PMap}(Σ)$ acts by permuting vertices. We prove that any connected, Gromov-hyperbolic cocompact arc and curve model $\mathcal{M}$ has $\operatorname{asdim} \mathcal{M} \geq g - \lceil\frac{1}{2} χ(Σ)\rceil$, and that a broad class of arc and curve models on infinite-type surfaces has infinite asymptotic dimension.

math.GT↗

arXiv2024

Automorphisms of the sphere complex of an infinite graph

For a locally finite, connected graph $Γ$, let $\operatorname{Map}(Γ)$ denote the group of proper homotopy equivalences of $Γ$ up to proper homotopy. Excluding sporadic cases, we show $\operatorname{Aut}(S(M_Γ)) \cong \operatorname{Map}(Γ)$, where $\mathcal{S}(M_Γ)$ is the sphere complex of the doubled handlebody $M_Γ$ associated to $Γ$. We also construct an exhaustion of $S(M_Γ)$ by finite strongly rigid sets when $Γ$ has finite rank and finitely many rays, and an appropriate generalization otherwise.

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arXiv2024

Multiarc and curve graphs are hierarchically hyperbolic

A multiarc and curve graph is a simplicial graph whose vertices are arc and curve systems on a compact, connected, orientable surface S. We show that all connected, non-trivial multiarc and curve graphs preserved by the natural action of PMod(S) and whose adjacent vertices have bounded geometric intersection number are hierarchically hyperbolic spaces with respect to witness subsurface projection. This result extends work of Kate Vokes on twist-free multicurve graphs and confirms two conjectures of Saul Schleimer in a broad setting. In addition, we prove that the PMod(S)-equivariant quasi-isometry type of such a graph is uniquely classified by its set of connected witness subsurfaces.

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arXiv2023

Prescribed Arc Graphs

Given a compact surface $Σ$ with boundary and a relation $Γ$ on $π_0(\partialΣ)$, we define the prescribed arc graph $\mathscr A(Σ,Γ)$ to be the full subgraph of the arc graph $\mathscr A(Σ)$ containing only classes of arcs between boundary components in $Γ$. We prove that $\mathscr(Σ,Γ)$ is connected and infinite-diameter (if $Σ$ is not the sphere with three boundary components), and classify when it is Gromov hyperbolic: in particular, $\mathscr(Σ,Γ)$ is Gromov hyperbolic if and only if $Γ$ is not bipartite, except in some sporadic cases.

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