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.

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 paperA Birman exact sequence for $\text{Aut}(F_n)$. $\endgroup$ – Andy Putman Sep 20 '13 at 16:20