Subset currents on surfaces

TL;DR

This paper develops the theory of subset currents on hyperbolic surface groups, generalizing geodesic currents, and introduces a measure-theoretic framework that extends intersection numbers to a continuous bilinear functional.

Contribution

It extends the concept of subset currents from free groups to surface groups, establishing a measure-theoretic completion and generalizing intersection theory.

Findings

The space of subset currents is a measure-theoretic completion of conjugacy classes of finitely generated subgroups.

Extension of intersection numbers to convex cores and a continuous bilinear functional.

Generalization of previous results from free groups and hyperbolic groups to surface groups.

Abstract

Subset currents on hyperbolic groups were introduced by Kapovich and Nagnibeda as a generalization of geodesic currents on hyperbolic groups, which were introduced by Bonahon and have been successfully studied in the case of the fundamental group $\pi_1 (\Sigma)$ of a compact hyperbolic surface $\Sigma$. Kapovich and Nagnibeda particularly studied subset currents on free groups. In this article, we develop the theory of subset currents on $\pi_1(\Sigma )$, which we call subset currents on $\Sigma$. We prove that the space $\mathrm{SC}(\Sigma)$ of subset currents on $\Sigma$ is a measure-theoretic completion of the set of conjugacy classes of non-trivial finitely generated subgroups of $\pi_1 (\Sigma )$, each of which geometrically corresponds to a convex core of a covering space of $\Sigma$. This result was proved by Kapovich-Nagnibeda in the case of free groups, and is also a generalization of Bonahon's result on geodesic currents on hyperbolic groups. We will also generalize several other results of them. Especially, we extend the (geometric) intersection number of two closed geodesics on $\Sigma$ to the intersection number of two convex cores on $\Sigma $ and, in addition, to a continuous $\mathbb{R}_{\geq 0}$-bilinear functional on $\mathrm{SC}(\Sigma)$.