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Geometric intersection number and analogues of the curve complex for free groups. (English) Zbl 1194.20046
From the authors’ summary: “For the free group $$F_N$$ of finite rank $$N\geq 2$$ we construct a canonical Bonahon-type, continuous and $$\text{Out}(F_N)$$-invariant geometric intersection form $$\langle\,,\,\rangle\colon\overline{\text{cv}}(F_N)\times\text{Curr}(F_N)\to\mathbb{R}_{\geq 0}$$. Here $$\overline{\text{cv}}(F_N)$$ is the closure of unprojectivized Culler-Vogtmann Outer space $$\text{cv}(F_N)$$ in the equivariant Gromov-Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that $$\overline{\text{cv}}(F_N)$$ consists of all very small minimal isometric actions of $$F_N$$ on $$\mathbb{R}$$-trees. The projectivization of $$\overline{\text{cv}}(F_N)$$ provides a free group analogue of Thurston’s compactification of Teichmüller space.”
To be precise the authors prove: Let $$N\geq 2$$. There exists a unique continuous map $$\langle\,,\,\rangle\colon\overline{\text{cv}}(F_N)\times\text{Curr}(F_N)\to\mathbb{R}_{\geq 0}$$ which is $$\mathbb{R}_{\geq 0}$$-homogeneous in the first argument, $$\mathbb{R}_{\geq 0}$$-linear in the second argument, $$\text{Out}(F_N)$$-invariant, and such that for every $$T\in\overline{\text{cv}}(F_N)$$ and in every $$g\in F_N\setminus\{1\}$$ we have $$\langle T,\eta_g\rangle=\|g\|_T$$.
Let $$N\geq 3$$. Then the graphs $$\mathcal T_0(F_N)$$, $$\mathcal F(F_N)$$, $$\mathcal F^*(F_N)$$ have infinite diameter.
Particular interest present sections 7 and 8.
The reference list contains 38 items.

##### MSC:
 20F65 Geometric group theory 20E05 Free nonabelian groups 37E25 Dynamical systems involving maps of trees and graphs 57M07 Topological methods in group theory
##### Keywords:
free groups; outer space; geodesic currents; curve complexes
Full Text:
##### References:
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