A couple of years ago, I wrote about virtually geometric words, in connection with finding surface subgroups of certain hyperbolic groups. A conjugacy class, represented by an element w, in a non-abelian free group F is called geometric if we can write F as the fundamental group of a handlebody H in such a way that w is represented by an embedded curve on the boundary. (More generally, everything here will apply just as well to finite sets of conjugacy classes.) The word w is virtually geometric if the handlebody H has a finite-sheeted cover in which the total preimage of a submanifold representing w is homotopic to an embedded multicurve on the boundary. Cameron Gordon and I were surprised to discover that there are words that are virtually geometric but not geometric .
Example: The Baumslag–Solitar relators (for coprime) are virtually geometric but not geometric.
Prompted by this discovery, we asked (perhaps rather hopefully) whether every word is virtually geometric.
Even before our paper was published, Jason Manning had answered our question by finding examples of words that are not virtually geometric . More recently, Chris Cashen has explained exactly when a word is virtually geometric , building on work of Jean-Pierre Otal  , as well as on joint work that Chris did with Natasa Macura . I’d like to take some time to explain the solution, which turns out to be really beautiful.
First, I should comment that, in a sense, we do understand exactly when a word is geometric. Indeed, there’s an algorithm due to Zieschang , which is nicely explained in some notes of Berge . The point is that w is geometric if and only if its minimal Whitehead graph, which you can find using Whitehead’s algorithm, is planar and satisfies some simple conditions. But of course this doesn’t help us with the property of being virtually geometric, as there are infinitely many finite-sheeted covers to check.
It’s natural to make some hypotheses about w. First of all, we should assume that F does not split freely relative to w. Topologically, this means that w is disc-busting, ie intersects every properly embedded essential disc in H. It can also be useful to assume that w is rigid, meaning that F does not split over a cyclic subgroup relative to w.
Otal used w to define a certain space, which encodes a lot of interesting information about w. Here’s one way to define it. Consider the Cayley graph of the free group F. It can be compactified by adding the set of ends of F, which is naturally topologised as a Cantor set, denoted by . The lifts of the conjugates of w in the Cayley graph form a set of lines in the Cayley graph, each of which stretches from one end to another. Otal’s decomposition space is obtained from by identifying pairs of points that are joined by a line. In particular, admits a two-to-one map from a Cantor set, and so is a one-dimensional compactum.
The decomposition space can be thought of as a sort of ‘Whitehead graph at infinity’ for w (this point of view is at the heart of Cashen and Macura’s paper). Like the Whitehead graph, it encodes information about splittings of F relative to w. For instance, Otal proved the following.
Theorem(Otal): F splits freely relative to w if and only if is disconnected.
The decomposition space is more canonical than a Whitehead graph—the boundary of the Cayley graph is a quasi-isometry invariant, and so doesn’t change if we change the generating set of F. In fact, it is precisely the Bowditch boundary, when F is thought of as a hyperbolic group relative to .
Otal also proved that the decomposition space encodes whether or not a word is geometric. Indeed, he proved:
Theorem(Otal): Suppose is rigid. Then the word w is geometric if and only if is planar.
Using this, we instantly get some insight into virtually geometric words: the decompositions space is unchanged when we pass to a finite-index subgroup, and hence, if w is rigid, then it is virtually geometric if and only if it is geometric.
Of course, many words are not rigid. Indeed, the Baumslag–Solitar words above are not. At the other extreme, F could be the fundamental group of a compact surface, and w might represent the boundary of that surface. In the case of a 3-punctured sphere, this is also rigid, but any other surface admits a large number of cyclic splittings corresponding to simple closed curves that are not boundary parallel. So surfaces can be thought of as maximally non-rigid. The surface case can also be characterised by the decomposition space.
Theorem(Otal): The space is homeomorphic to a circle if and only if F is isomorphic to the fundamental group of a compact surface in such a way that w represents the boundary.
Non-rigidity is also reflected in the decomposition space.
Lemma(Otal): If F splits over a cyclic subgroup relative to w then has either a cut point or a cut pair.
Building on his work with Macura, Cashen proved that the converse holds, and indeed that the decomposition space encodes the JSJ decomposition of F relative to w—that is, in a sense, the decomposition space sees every cyclic splitting of F relative to w. (This result is very similar to Bowditch’s theorem that the boundary of a word-hyperbolic group encodes the JSJ decomposition of that group .)
As in the case of a word-hyperbolic group, the relative JSJ decomposition of F is a graph of groups with two sorts of vertices: rigid vertices, just as above, admit no cyclic splittings relative to their incident edge groups or w. The remaining vertex groups can be thought of as the fundamental groups of surfaces, with the conjugates of w and the incident edge groups attached along boundary components.
Theorem(Cashen): Let w be a word in F such that the corresponding decomposition space is connected. There is a canonical graph of groups decomposition of F relative to w with the following properties:
- The vertex groups are free (possibly cyclic), and any edge groups are cyclic.
- In every non-cyclic vertex group G, the decomposition space of G generated by stabilizers of the incident edge groups and conjugates of w either is a circle or is rigid.
- Either is rigid or a circle or the decomposition is nontrivial.
The graph of groups is non-trivial if and only if is not a circle and contains either a cut point or a cut pair.
From this, Cashen deduces that there are precisely two obstructions to w’s being geometric.
- If some vertex group is rigid and non-geometric then w is not geometric.
- If the image of some attaching map of some edge group admits a proper root then w is not geometric.
The point about the second obstruction is that a proper power of a curve on the surface of a handlebody is not itself embedded on the surface of the handlebody. But this second obstruction can always be removed by passing to a finite-index subgroup, using Marshall Hall’s Theorem. Indeed, this is what is going on in the Baumslag–Solitar example. A non-geometric word exhibiting only the second obstruction will be virtually geometric. However, as we saw above, the first obstruction cannot be resolved by passing to a finite-index subgroup.
Finally, I should mention that Cashen and Macura exhibited an algorithm that computes cyclic splittings of F relative to w, which can be used to determine the JSJ decomposition of F, and hence whether or not a word is virtually geometric. In fact, an earlier algorithm, that works in a much more general context, was described by Dahmani and Groves . However, Dahmani and Groves’ algorithm probably has non-computable running time. Cashen and Macura’s algorithm is certainly much quicker, though I don’t know if it is quick enough to actually be implemented.
 Cameron Gordon, Henry Wilton, On surface subgroups of doubles of free groups. J. Lond. Math. Soc. (2) 82 (2010), no. 1, 17–31.
 Jason Fox Manning, Virtually geometric words and Whitehead’s algorithm. Math. Res. Lett. 17 (2010), no. 5, 917–925.
 Christopher H. Cashen, Splitting line patterns in free groups, arXiv:1009.2492v2 [math.GR]
 Jean-Pierre Otal, Certaines relations d’équivalence sur l’ensemble des bouts d’un groupe libre. (French) [Some equivalence relations in the set of ends of a free group] J. London Math. Soc. (2) 46 (1992), no. 1, 123–139.
 Christopher H. Cashen, Natasa Macura, Line patterns in free groups, arXiv:1006.2123v4 [math.GR]
 H. Cišang, Simple path systems on full pretzels. (Russian) Mat. Sb. (N.S.) 66 (108), 1965, 230–239.
 John Berge, Heegaard documentation, http://www.math.uic.edu/~t3m/hg/heegaard/Documentation/Heegaard_Documentation.pdf
 Brian H. Bowditch, Cut points and canonical splittings of hyperbolic groups. Acta Math. 180 (1998), no. 2, 145–186.
 François Dahmani, Daniel Groves, The isomorphism problem for toral relatively hyperbolic groups. Publ. Math. Inst. Hautes Études Sci. No. 107 (2008), 211–290.