发表机构
Hunter College of CUNY（纽约市立大学亨特学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在自由群和曲面群上构造了具有真正分支支撑集的非有理子集流，证明其支撑集可为康托尔集，并揭示Choquet单纯形作为流锥体基的普适性，同时研究了有限模式复杂性与锥体维数的关系。
AI 中文摘要
子集流（subset currents）在自由群、曲面群以及更一般的字双曲群上由Kapovich和Nagnibeda引入，并由Sasaki进一步研究。子集流推广了普通的测地流，并为无限拟凸子群的共轭类提供了测度论上的推广。它们的支撑集可以被视为“分支叶状结构”（branching laminations）。对于自由群和曲面群，我们构造了具有真正分支支撑集的非有理子集流，这些流除了标量倍数外不携带其他子集流。对于每个非交换有限秩自由群和每个闭双曲曲面群，我们证明这样的流可以被选择为使其支撑集是不可数的，与两点边界子集轨迹不相交，并且完全由边界的康托尔子集组成。更一般地，我们证明每个非空可度量化Choquet单纯形都可以作为由极小真正分支子集叶状结构承载的流的锥体的紧凸基。同样的普适性现象，除了通常的有理流术语外，也扩展到每个非初等无挠双曲群。我们还研究了有限模式复杂性。我们证明，如果$R_L(n)$计数极小子集叶状结构$L$中半径为$n$的允许“圆形模式”（round patterns），那么$R_L(n)=O(n)$迫使承载流的锥体是有限维的，而无限维性迫使$R_L(n)/n\to\infty$；这个普适阈值是精确的。我们证明了$R_L$的粗略增长类型独立于所选的自由基，更一般地，独立于所选的有限标记图。我们还表明，一个精细的开关系统可以恢复承载流的锥体的精确维数。
英文摘要
\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.
Comments34 pages
