an:05680962
Zbl 1242.20052
Kapovich, Ilya; Lustig, Martin
Intersection form, laminations and currents on free groups
EN
Geom. Funct. Anal. 19(2009), No. 5, 1426-1467 (2010).
00258808
2010
j
20F65 20E05 20F28 20E08 57M07 37A35 37B10 37E25
free groups; outer space; geodesic currents; geometric intersection numbers; isometric actions; translation length functions; algebraic laminations; filling elements
Summary: Let \(F\) be a free group of rank \(N\geq 2\), let \(\mu\) be a geodesic current on \(F\) and let \(T\) be an \(\mathbb R\)-tree with a very small isometric action of \(F\). We prove that the geometric intersection number \(\langle T,\mu\rangle\) 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 \textit{S. Francaviglia} [Trans. Am. Math. Soc. 361, No. 1, 161-176 (2009; Zbl 1166.20032)] regarding length spectrum compactness for currents with full support. We use the main result to obtain ``unique ergodicity'' type properties for the attracting and repelling fixed points of atoroidal iwip elements of \(\text{Out}(F)\) when acting both on the compactified outer space and on the projectivized space of currents. We also show that the sum of the translation length functions of any two ``sufficiently transverse'' very small \(F\)-trees is bilipschitz equivalent to the translation length function of an interior point of the outer space. As another application, we define the notion of a filling element in \(F\) and prove that filling elements are ``nearly generic'' in \(F\). We also apply our results to the notion of bounded translation equivalence in free groups.
Zbl 1166.20032