Geometric Intersection Number and analogues of the Curve Complex for free groups

By Ilya Kapovich, Martin Lustig. 2007

109 citations recorded by the source corpus.

For the free group $F_{N}$ of finite rank $N \geq 2$ we construct a canonical Bonahon-type continuous and $Out(F_N)$-invariant \emph{geometric intersection form} \[ : \bar{cv}(F_N)\times Curr(F_N)\to \mathbb R_{\ge 0}. \] Here $\bar{cv}(F_N)$ is the closure of unprojectivized Culler-Vogtmann's Outer space $cv(F_N)$ in the equivariant Gromov-Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that $\bar{cv}(F_N)$ consists of all \emph{very small} minimal isometric actions of $F_N$ on $\mathbb R$-trees. The projectivization of $\bar{cv}(F_N)$ provides a free group analogue of Thurston's compactification of the Teichm\"uller space. As an application, using the \emph{intersection graph} determined by the intersection form, we show that several natural analogues of the curve complex in the free group context have infinite diameter.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f636f570c5241868ffdafde09b2e29df4c6ee53c

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Citations in the corpus

Hyperbolicity of the complex of free factors (2011, background): “It is also known that fully irreducible automorphisms act on F with positive translation length [23, 5].”
The boundary of the complex of free factors (2012, background): “an element of the form hk for h ∈ FN and k > 1, then there is a counting current, denoted ηg, associated to the conjugacy class of g. We also set ηgk = kηg and frequently write g instead of ηg. In [KL09], Kapovich and Lustig establish the following: Proposition2.1.[KL09, Theorem A] There is a uni”
On hyperbolicity of free splitting and free factor complexes (2012, background): “Several years ago Kapovich-Lustig [18] and Behrstock-Bestvina-Clay [2] showed that for N ≥ 3 the (quasi-isometric) complexes FFN and IN have infinite diameter.”
The co‐surface graph and the geometry of hyperbolic free group extensions (2016, background): “It is natural to build a better-suited complex by starting with F and coning off the curve graphs for all once-punctured surfaces S with π1(S) ∼= F. Versions of this construction appear several places in the literature — first in the work of Kapovich–Lustig [28] and Mann–Reynolds (private communicat”
A hyperbolic Out(F_n)-complex (2008, background): “erbolic group [Bon91]. Of interest for this paper is the case of free groups, further studied by Reiner Martin in his thesis [Mar95], and more recently by Ilya Kapovich, Martin Lustig and others (see [KL09] and references therein). Martin’s thesis has never been published, but most of his results ar”
Hyperbolic extensions of free groups (2014, methodology): “We begin by deﬁning a version of the intersection graph I for F , which is an Out ( F ) –graph introduced by Kapovich and Lustig in [KL1].”