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Let $F$ be a finitely generated free group.

Question: Is there an authoritative survey of analogues of the curve complex for $\mathrm{Out}(F)$? If not, as seems likely, would a passing expert be willing to write a brief summary answering the following questions.

  • Which analogues are thought most important?
  • Why?
  • How do they relate to each other?
  • What's known about them? (eg are they of infinite-diameter, contractible, Gromov-hyperbolic etc?)

Background

A curve complex $\mathcal{C}(\Sigma)$ is a certain simplicial complex associated to a compact surface $\Sigma$, first defined by Bill Harvey. The definition is simple and well known: the vertices are simple curves on $\Sigma$, and curves $(\gamma_i)$ span a simplex if and only if they're mutually disjoint. They turn out to be very useful when studying mapping class groups $\mathrm{Mod}(\Sigma)$, and are an area of intensive research. By the way, all the information is actually carried by the 1-skeleton, the curve graph.

(I'm a little shocked to see that there's no wikipedia article on curve complexes.)

In analogy with $\mathrm{Mod}(\Sigma)$, we are also interested in the group of outer automorphisms $\mathrm{Out}(F)$, where $F$ is a free group, and a nice strategy is to define analogous objects in this context. For instance, Culler–Vogtmann Outer Space is the analogue of Teichmüller space etc.

Of course, analogies aren't always perfect, and sometimes on object in one context may have several analogues in another context. In this case, this problem seems to have occurred many times over! There is a bewildering variety of analogues of the curve complex and the curve graph for $\mathrm{Out}(F)$. A brief literature search found the following:

  • The free splitting complex (Handel--Mosher)
  • The free factor complex (Hatcher--Vogtmann)
  • The dual free splitting graph (Kapovich--Lustig)
  • The sphere complex (Hatcher)
  • The cyclic splitting complex (Mann)
  • The edge splitting graph (Sabalka--Savchuk)
  • The primitivity graph (Kapovich--Lustig)
  • The cut graph (Kapovich--Lustig)
  • The dual cut graph (Kapovich--Lustig)
  • The ellipticity graph (Kapovich--Lustig)
  • The dominance graph (Kapovich--Lustig)

The names just indicate people who have worked on these - they're not meant to be authoritative.

Although this question may admit multiple answers, I'd like to resist making it community wiki. The accepted answer should either be a reference to a survey, or a fairly detailed summary of the leading analogues, their relationship to each other, and what's known about them. An answer of the latter sort would require a fair bit of work, and so would deserve credit.

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    $\begingroup$ Can I add to the list? The graph whose vertices are conjugacy classes of rank 1 factors and whose edges correspond to pairs of vertices represented by elements that generate a rank 2 factor is quasi-isometric to the factor complex (as are others in your list) but has local properties similar to the curve complex. For example, this graph interacts nicely with subfactor projections. $\endgroup$ – staylor Sep 20 '13 at 14:40
  • $\begingroup$ @staylor, Yes! The list is not meant to be exhaustive - just to indicate the scale of the problem for a non-expert interested in the area. An account of which complexes are known to be quasi-isometric to each other is exactly the sort of information that I'd be interested in. $\endgroup$ – HJRW Sep 20 '13 at 14:48
  • $\begingroup$ @staylor : That space is useful in other contexts too. For instance, Matt Day and I used a version of it to study the Torelli subgroup of $\text{Aut}(F_n)$ in our paper The complex of partial bases for $F_n$ and finite generation of the Torelli subgroup of $\text{Aut}(F_n)$ and gave descriptions of the stabilizers of simplices in it in our paper A Birman exact sequence for $\text{Aut}(F_n)$. $\endgroup$ – Andy Putman Sep 20 '13 at 16:20
  • $\begingroup$ It's misleading to say that "all the information" about the curve complex is contained in its one-skeleton. As shown by Harer, the homotopy type of the curve complex is a wedge of spheres, and this is used in establishing the v.c.d. of the mapping class group. $\endgroup$ – Lee Mosher Sep 22 '13 at 3:29
  • $\begingroup$ @LeeMosher - I was referring to the fact that the curve complex is flag and so can be reconstructed easily from the curve graph. I wanted to explain the fact that some of the free-group analogues are just graphs. Next time I edit the question, I'll try to make this clearer. $\endgroup$ – HJRW Sep 22 '13 at 6:36
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I'll take a crack at this (this will be preliminary; I will correct if, as I expect, I forget things).

There is no survey that I know of. Knowledge is moving fast on this topic. It is perhaps too early to stretch too far at guessing which complexes will be important and which will not be, so I will stick to a more descriptive overview of what has been proved, focussing on hyperbolicity and other properties in the question.

The complexes known to be hyperbolic are:

  • The free factor complex ($\approx$ several others, including some on the Kapovich-Lustig list). Proof by Bestvina and Feighn.
  • The free splitting complex ($\approx$ sphere complex). Proof by Handel and me.
  • The cyclic splitting complex. Proof by Brian Mann.

The original hyperbolicity proof for the free splitting complex was expressed in the language of $F_n$ actions on trees. Hilion and Horbez reworked that proof in the language of Hatcher's sphere systems, introducing some simplifications. Bestvina and Feighn reworked the proof again, back in the language of $F_n$ actions on trees, introducing other simplifications. A major effect of these different proofs is to exhibit different classes of reparameterized quasigeodesics in the free splitting complex. In the case of the free splitting complex, the class of quasigeodesics called "fold paths" is given an explicit quasigeodesic parameterization.

Important relations amongst these complexes are natural equivariant Lipschitz maps from the free splitting complex to both the free factor complex and the cyclic splitting complex. Kapovich and Rafi exploited the first of these maps to give a new proof of hyperbolicity of the free factor complex, deriving it from hyperblicity of the free splitting complex. Mann subsequently used the same method in proving hyperbolicity of the cyclic splitting complex.

The $Out(F_n)$ action on each of the above hyperbolic complexes is known to contain loxodromic elements (i.e. having a quasi-axis), and so in particular these complexes are all of infinite diameter. In general the subset of $Out(F_n)$ acting loxodromically on the free splitting complex contains the sets acting loxodromically on the free factor and cyclic splitting complexes, because of the existence of an $Out(F_n)$-equivariant Lipschitz map from the free splitting complex to the other two. Also, the loxodromic sets for these three complexes are pairwise distinct, hence none of these complexes is $Out(F_n)$-equivariantly isomorphic to the other, nor even equivariantly quasi-isometric. Here are some details.

  • In the free factor complex, the loxodromic elements are the fully irreducible outer automorphisms (see Bestvina-Handel).
  • In free splitting complex, the loxodromic elements form a strictly larger class, namely those outer automorphisms having an attracting lamination that fills $F_n$ (Handel lectured on this in summer 2013 in Oberwohlfach; we hope to post this on the arXiv "soon").
  • In the cyclic splitting complex, Mann describes outer automorphisms that are loxodromic here but not in the free factor complex.

The edge-splitting complex of $F_n$ was proved by Sabalka and Savchuk to be non-hyperbolic, by showing that it contains quasiflats of arbitrarily high dimension. There is an analogue to this in Schleimer's proof that the separating curve complex of a surface is not hyperbolic, again with flats but not of arbitrarily high dimension.

As for topology/homotopy theory, here's what I know about:

  • The free factor complex of $F_n$ is homotopy equivalent to a wedge of spheres of dimension $n-2$. Proof by Hatcher and Vogtmann (MR1660045 (99i:20038)).
  • The sphere complex ($\approx$ free splitting complex) is contractible. Proof by Hatcher (MR1314940 (95k:20030)).
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  • $\begingroup$ Thanks, for this, Lee. It's pretty much exactly what I was hoping for. I take it $\approx$ means q.i.? $\endgroup$ – HJRW Sep 24 '13 at 12:08
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    $\begingroup$ Actually, it's a big ambiguous. In the case of the free splitting complex and the sphere complex it means equivariantly simplicially isomorphic. But in the case of the free factor complex and various others, it does only mean equivariantly quasi-isometric; there seem to be several useful ones, for instance see the Kapovich-Rafi paper. $\endgroup$ – Lee Mosher Sep 25 '13 at 4:36
  • $\begingroup$ Small correction: the free factor complex is homotopy equivalent to a wedge of spheres. $\endgroup$ – HJRW Sep 25 '13 at 4:48
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    $\begingroup$ Oops, fixed. I always have to pause and say "the free....... FACTOR complex" or "the free...... SPLITTING complex", and I still get it wrong sometimes. $\endgroup$ – Lee Mosher Sep 25 '13 at 13:23

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