Intersection form, laminations and currents on free groups

Details Intersection form, laminations and currents on free groups Intersection form, laminations and currents on free groups

2007-11-27

arxiv.org/abs/0711.4337v2

Geom. Funct. Anal. vol. 19 (2010), no. 5, pp. 1426-1467

Mathematics

Geometric Topology

revised version, to appear in GAFA

Scientific paper

10.1007/s00039-009-0041-3

Let $F_N$ be a free group of rank $N\ge 2$, let $\mu$ be a geodesic current on $F_N$ and let $T$ be an $\mathbb R$-tree with a very small isometric action of $F_N$. We prove that the geometric intersection number $$ is equal to zero if and only if the support of $\mu$ is contained in the dual algebraic lamination $L^2(T)$ of $T$. Applying this result, we obtain a generalization of a theorem of Francaviglia regarding length spectrum compactness for currents with full support. As another application, we define the notion of a \emph{filling} element in $F_N$ and prove that filling elements are "nearly generic" in $F_N$. We also apply our results to the notion of \emph{bounded translation equivalence} in free groups.

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