Uniquely Ergodic Branching Subset Currents on Free and Surface Groups

Abstract

\emph{Subset currents} on free groups, on surface groups and, more generally, on word-hyperbolic groups, were introduced by Kapovich and Nagnibeda, and further studied by Sasaki. Subset currents extend ordinary geodesic currents and provide measure-theoretic generalizations of conjugacy classes of infinite quasiconvex subgroups. Their supports can be viewed as ``branching laminations". For free groups and surface groups, we construct non-rational subset currents with genuinely branching supports that carry no other subset currents except scalar multiples. For every nonabelian finite rank free group and every closed hyperbolic surface group, we show that such a current can be chosen so that its support is uncountable, is disjoint from the locus of two-point boundary subsets, and consists entirely of Cantor subsets of the boundary. More generally, we prove that every nonempty metrizable Choquet simplex occurs as a compact convex base for the cone of currents carried by a minimal genuinely branching subset lamination. The same universality phenomenon extends, apart from the usual rational-current terminology, to every non-elementary torsion-free hyperbolic group. We also study finite-pattern complexity. We prove that if $R_L(n)$ counts the allowed \emph{round patterns} of radius $n$ in a minimal subset lamination $L$, then $R_L(n)=O(n)$ forces the carried-current cone to be finite-dimensional, while infinite-dimensionality forces $R_L(n)/n\to\infty$; this universal threshold is sharp. We show that the coarse growth type of $R_L$ is independent of the chosen free basis and, more generally, of the chosen finite marked graph. We also show that a refined switch system recovers the exact dimension of the carried-current cone.