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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
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