Infinite-type loxodromic isometries of the relative arc graph

TL;DR

The authors construct an infinite family of intrinsically infinite-type mapping classes $\{g_n\}$ on infinite-type surfaces with an isolated puncture, each acting loxodromically on the relative arc graph $\mathcal A(\Sigma,p)$. Each $g_n$ is realized as a composition of three shift maps on a biinfinite flute model, with an alternate description as a finite-type pseudo-Anosov on a subsurface followed by a shift, and they extend these elements to surfaces $\Sigma$ of type $\mathcal S$. They identify quasi-geodesic axes for $g_n$ in $\mathcal A(\Sigma,p)$, determine their limit points on the Gromov boundary, and demonstrate a convergence to a geodesic lamination carried by a train track, thereby connecting dynamics with laminations. The paper further shows that these intrinsically infinite-type elements yield an infinite-dimensional space of quasimorphisms on $\mathrm{Map}(\Sigma,p)$, even though the elements are not WWPD. Collectively, the results advance a Nielsen–Thurston-type understanding for infinite-type mapping classes via hyperbolic geometry and train-track techniques, and they broaden the known repertoire of loxodromic, intrinsically infinite-type elements beyond the plane minus a Cantor set.

Abstract

An infinite-type surface $Σ$ is of type $\mathcal{S}$ if it has an isolated puncture $p$ and admits shift maps. This includes all infinite-type surfaces with an isolated puncture outside of two sporadic classes. Given such a surface, we construct an infinite family of intrinsically infinite-type mapping classes that act loxodromically on the relative arc graph $\mathcal{A}(Σ, p)$. J. Bavard produced such an element for the plane minus a Cantor set, and our result gives the first examples of such mapping classes for all other surfaces of type $\mathcal{S}$. The elements we construct are the composition of three shift maps on $Σ$, and we give an alternate characterization of these elements as a composition of a pseudo-Anosov on a finite-type subsurface of $Σ$ and a standard shift map. We then explicitly find their limit points on the boundary of $\mathcal{A}(Σ,p)$ and their limiting geodesic laminations. Finally, we show that these infinite-type elements can be used to prove that Map$(Σ,p)$ has an infinite-dimensional space of quasimorphisms.

Figures (53)

• Figure 1: A handleshift on an infinite-type surface (above) and a shift map on an infinite-type surface (below).
• Figure 2: Examples of sporadic infinite-type surfaces that are not of type $\mathcal{S}$. The first is a flute with finite genus, the other three are fluted Loch Ness Monster surfaces.
• Figure 3: Examples of surfaces of type $\mathcal{S}$, the first of which is the biinfinite flute surface $S$ itself.
• Figure 4: Examples showing that surfaces with shift maps are always of type $\mathcal{S}$. For the first surface, only one curve $\gamma$ is needed to cut away the extra topology of $\Sigma$. In the second case, a countable collection of curves $\{\gamma_i\}$ is needed.
• Figure 5: The curves $B_i$ are in red. The blue region is the domain of the shift map $H$.

Theorems & Definitions (125)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6