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Jeremy Kahn kindly sent me a more detailed overview of his argument with Vlad Markovic, that I blogged earlier about here (also see Jesse Johnson’s blog for other commentary). With his permission, this is reproduced below in its entirety.

Editorial note: I have latexified Jeremy’s email; hence “dhat-mu” becomes $\hat{d}\mu$, “boundary-hat” becomes $\hat{d}$, and “boundary-tilde” becomes $\tilde{d}$. I also linkified the link to Caroline Series’ paper.

Hi Danny,

I was busy with the conference on Thursday and Friday, and taking a break on Saturday, and now I’ve finally had a chance to read your blog, and reply to your message. I decided (especially as Jesse had requested it) to write out a complete outline of the theorem. I’m sending a copy of this message to you, Jesse Johnson, Ian Agol, and Francois Labourie: you are all welcome to reproduce it, as long as it is reproduced in its entirety, and states clearly that this is joint work with Vladimir Markovic. Of course, time and energy permitting, I’ll be happy to answer any questions.

Here is an outline of the argument, working backwards to make it clearer:

1. We want to construct a surface made out of skew pants, each of which has complex half-length close to $R$, and which are joined together so that the complex twist-bends are within $o(1/R)$ of $1$. Using a paper of Caroline
Series (published in the Pacific J. of Mathematics) we show that these surfaces are quasi-isometrically embedded in the universal cover of the three-manifold.

2. Consider the following two conditions on two Borel measures $\mu$ and $\nu$ on a metric space $X$ with the same (finite) total measure:

A. For every Borel subset $A$ of $X$, $\mu(A)$ is less than or equal to the $\nu$-measure of an $\epsilon$ neighborhood of $A$.

B. There is a measure space $(Y, \eta)$ and functions $f: Y \to X$ and $g: Y \to X$ such that $\mu$ and $\nu$ are the push-forwards by $f$ and $g$ respectively of the measure $\eta$, and the distance in $X$ between $f(y)$ and $g(y)$ is less than $\epsilon$ for almost every $y \in Y$.

It is easy to show that B implies A (also that A is symmetric in $\mu$ and $\nu$!). In the case where $\mu$ and $\nu$ are discrete and integral measures (the measure of every point is a non-negative integer), we can show that A implies B (and $Y$ will be a finite set with the counting measure) using Hall’s marriage theorem. In fact, the statement that A implies B for discrete and integral measures is easily shown to be equivalent to Hall’s marriage theorem. I don’t know if A implies B in general because I don’t know how to replace the inductive algorithm for Hall’s marriage theorem with a method that works for a relation between two general measure spaces.

We call $\mu$ and $\nu$ $\epsilon$-equivalent if they satisfy condition A, and note that the condition is additively transitive: if $\mu$ is $\epsilon$-equivalent to $\nu$, and $\nu$ is $\delta$-equivalent to $\rho$, then $\mu$ and $\rho$ are $(\epsilon+\delta)$-equivalent.

3. Suppose that $\gamma$ is one boundary component of a pair of skew pants $P$. We can form the common orthogonals in $P$ from $\gamma$ to each of other other two cuffs. For each common orthogonal, at the point where it meets $\gamma$, we can find a unit normal vector to $\gamma$ that points along this common orthogonal. The two resulting normal vectors are related by a translation along the half-length of $\gamma$ (the suitable square root of the loxodromic element for $\gamma$), so we will call them a pair of opposite unit normal vectors (or pounv for short) and they live in the live in the bundle of pounv’s which is conformally equivalent to the complex plane mod the lattice generated by the half-length of $\gamma$ and $2\pi i$. We give the bundle of pounv’s the Euclidean metric inherited from the complex plane, and also the Lebesgue measure.

4. Given a measure on pants we can produce a measure on the union pounv bundles of the boundary geodesics as follows: if the measure is a unit atom on one pair of skew pants, the resulting measure on pounv bundles is a unit atom on the pounv bundle of each the cuffs, at the pounv described in step 3. We extend to a general measure by linearity. This produces a linear operator we will call the $\hat{d}$ operator.

If we are given a positive integral formal sum of pants (or a multi-set of pants) we can think of it as an integral measure on the space of pants.

5. On the pounv bundle for each closed geodesic we can apply a translation of $1 + i \pi$; we will call this translation $\tau$. We can think of $\tau$ as a map from the union of the pounv bundles to itself.

6. Let $\mu$ be an integral measure on pants with cuff half-lengths close to $R$. We can apply the $\hat{d}$ operator described in step 4 to obtain a measure on the union of pounv bundles of all the boundary geodesics; we will call the measure $\hat{d}\mu$. If $\hat{d}\mu$ and the translation of $\hat{d}\mu$ by $\tau$ are $\epsilon/R$ equivalent, then we can take two oriented pants for each pair of pants in our multi-set (taking each of the two possible orientations) and then fit all of these oriented pants into an oriented surface of the type described in step 1. We use Hall’s marriage theorem as described in step 2, and a very small amount of combinatorics.

If the measure $\hat{d}\mu$, restricted to a given pounv bundle, is $\epsilon/R$ equivalent to a rescaling of Lebesgue measure on that torus, then $\hat{d}\mu$ and $\tau$ of $\hat{d}\mu$ are $2\epsilon/R$-equivalent, which is what we wanted.

******************

This is as far as I got in the first talk at Utah, so it would be best to stop and take a breath for a moment. We haven’t really done anything, but we’ve reformulated the problem: the type of surface we want has been well-defined, and the problem of finding this surface has been reformulated as finding a measure on pairs of pants that satisfies a given criterion.

*****************

7. A two-frame for $M$ will comprise a tangent vector and a normal vector both at the same point, unit length and orthogonal. Given a two-frame we can rotate the tangent vector 120 degrees around the normal vector, using the right-hand rule; the orbit of this action is an ordered triple of two-frames, which will call a tripod. We can also rotate 120 degrees in the opposite direction, and obtain an anti-tripod.

8. A connected pair of two-frames is a pair of two frames along with a geodesic segment connecting them. Given $\epsilon$ and $r$, with $r$ large in terms of $\epsilon$, we can find a weighting function on connected two-frames such that the following properties hold whenever the weight is non-zero:

A. The length of the connecting segment is within $\epsilon$ of $r$.

B. If the normal vector of one two-frame is parallel translated along the connecting segment, then it forms an angle of less then $\epsilon$ with the normal vector of the other two-frame.

C. The angle between the the tangent vector of the two frame and (the tangent vector to) the connecting geodesic segment is exponentially small in $r$.

Moreover,

D. Given a pair of two-frames, the sum of the weights of the connecting geodesic segments is exponentially close (in $r$) to 1.

E. The weighting is geometrically natural, in that it depends only the length of the connecting segment, the angle between the parallel translated normal vectors, and the angles between the connecting segment and the tangent vectors.

We will describe the (relatively simple) weighting function in the end; we will use the exponential mixing of geodesic flow to obtain property D.

9. Given a tripod and an anti-tripod, we can form three pairs of two-frames by pairing the frames in order, and then we can measures (or weightings) on the connected pairs of two-frames, and then form the product measure (or weighting) by multiplying the weights of the three connections. This gives us a weighting on “connected pairs of tripods” (really a tripod and an anti-tripod) that is supported on connections that satisfy properties A, B, and C.

10. We call a perfect connection between two two-frames a geodesic segment that has a length of $r$, and angle of zero between the segment and the tangent vectors, and translates one normal vector to the other. If a tripod and an anti-tripod were connected by three perfect connection, then they would be a 1-dimensional retract of a flat pair of pants with three cuffs of equal length $R$, where $R$ is approximately $r + \log \cos \pi/6$ when $r$ is large. If the tripod and anti-tripod are connected by arcs that satisfy properties A and B, then the connected pair of tripods is still a retract of a skew pair of pants, whose cuffs have half-length within $\epsilon$ (or $10\epsilon$) of $R$. Thus there is a map from good connected pairs of tripods to good pairs of pants, which we will denote by $\pi$.

11. We can let $\tilde{\mu}$ be the measure on connected pairs of tripods, given by integrating the weighting of steps 8 and 9 with respect to the Liouville measure on pairs of tripods (or pairs of two-frames). We then push this measure forward by $\pi$ to obtain a measure $\mu$ on pairs of pants; after finding a rational approximation and clearing denominators, it will be the $\mu$ that was asked for in step 6. We will show that $\hat{d}\mu$ (taking the original irrational $\mu$) is $\epsilon/R$-equivalent to a rescaling of Lebesgue measure on each pounv bundle and thereby complete the proof.

12. A partially connected pair of tripods $T$ is a pair of tripods where we have connected two out of the three pairs of two-frames. To a partially connected pair of tripods we can assign a single closed geodesic $\gamma$ that is homotopic to the concatenation (at both ends) of the two connecting segments. If we connect the third pair of two-frames and apply $\pi$ we obtain a pair of pants $P$, and we can then find a pair of opposite unit normal vectors for gamma pointing to the two cuffs of $P$ (as described in step 3). We will describe a method for predicting the pounv for $\gamma$ and $P$ knowing only the partially connected tripod $T$: First, lift $T$ to the solid torus cover of $M$ determined by $\gamma$, and then follow geodesic segments from the tangent vectors of the two unconnected two frames of (the lift of) $T$ to the ideal boundary of this $\gamma$-cover. We can connect these two points in the boundary by two geodesics, each of which goes about half-way around this solid torus cover. We can then find the common orthogonals from each of these geodesics to (the lift of) $\gamma$, and then obtain two normal vectors to $\gamma$ pointing along these common orthogonals; it is easy to verify that these are half-way along $\gamma$ from each other (in the complex sense) and hence form a pounv. Property C of the connections between two-frames (and hence tripods) implies that this predicted pounv will be exponentially close (in $r$) to the actually pounv of any pair of pants $P$.

To summarize: given a good connected pair of tripods, we get a good pair of pants $P$, and taking one cuff gamma of $P$, we get a pounv for $\gamma$ as described in step 3. But we only need two out of the three connecting segments to get $\gamma$, and using the third pair of two frames, without even knowing the third connecting segment, we can predict the pounv for $\gamma$ and $P$ to very high accuracy.

13. We can then define the $\tilde{d}$ operator from measures on partially connected pairs of tripods to measures on the pounv bundles for the associated geodesics; this operator is just the linear extension of the operation in step 12. Given a connected pair of tripods, we can get three partially connected pairs of tripods in the obvious way; we can thereby extend $\tilde{d}$ to map measures on connected pairs of tripods to measures on the bundles of pounv’s; because the predicted pounv described in step 12 is exponentially close to the actual pounv described in step 3, the two measures $\tilde{d} \tilde{\mu}$ and $\hat{d}\mu$ are $\exp(-\alpha r)$-equivalent, by the B => A of step 2.

14. For each closed geodesic $\gamma$, we can lift all the partially connected tripods that give $\gamma$ to the $\gamma$ cover of $M$ described in step 12. There is a natural torus action on the normal bundle of $\gamma$, and this extends to an action on all of the solid torus cover associated to $\gamma$. Moreover, it acts on the (lifts of) partially connected tripods, and it does not change the weightings of the two established connecting segments, because of property E of the weighting function.

This is the crucial point: the effective weighting on a partially connected pair of tripods is not just the product of the weights of the two established connections, but that product times the sum of the weights of all possible third connections. By property D of the weighting function, this sum, while not constant, is exponentially close to being constant, so the effective weighting is exponentially close to being invariant under the torus action. Because the predicted pounv for a partially connected pair of tripods is equivariant for the torus action, the measure $\tilde{d} \tilde{\mu}$ is exponentially close to a torus invariant measure on the pounv bundle (which is necessary a rescaling of Lebesgue measure), in the sense that the Radon-Nikodym derivative is exponentially close to 1. It is then an easy lemma that the two measures are exponentially close in the sense of step 2. And then we’re finished: $\hat{d}\mu$ is exponentially close to $\tilde{d} \tilde{\mu}$, which is exponentially close to a rescaling of Lebesgue measure, which is what we wanted (with
overkill) in step 6.

15. It remains only to define the weighting function described in step 8, which is surprisingly simple: We take some left-invariant metric on $\text{PSL}_2(\bf{C})$, and hence on the two-frame bundle for $M$ and its universal cover. Given a connected pair of two-frames in $M$, we lift to the universal cover, to obtain two two-frames $v$ and $w$. We then flow $v$ and $w$ forward by the frame flow for time $r/4$ to obtain $v'$ and $w'$. We let $V$ be the $\epsilon$ neighborhood of $v'$, and $W$ be the $\epsilon$ neighborhood of $w'$, with the tangent vector of $w'$ replaced by its negation. Then the weighting of the connection is the volume of the intersection of $W$ with the image of $V$ under the frame flow for time $r/2$.

Properties A, B, and C are not difficult to verify. Property D follows immediately from exponential mixing: If we have $v$ and $w$ downstairs without any connection, and similarly define $v'$, $w'$, $V$ and $W$, then the sum of the weights of the possible connections will just be the volume of the intersection of the downstairs $W$ with the frame flow of $V$. By exponential mixing, this converges at the rate $\exp(-\alpha r)$ to the square of the volume of an $\epsilon$ neighborhood, divided by the volume of $M$.

We can normalize the weights by dividing by this constant.

Jeremy

One obvious comment to make is that the argument is remarkably short, and does not depend on any very delicate or complicated analytic estimates (maybe the argument that the glued up surfaces are quasi-geodesic is the most delicate part). It is fair to say that it defies the conventional wisdom in that respect — I was personally very surprised that the general method could be made to work, especially in light of the failure of Bowen’s program. Kudos to Jeremy and Vlad for their boldness and ingenuity.

Another comment to make is that the matching argument is surprisingly robust and general, and I expect it to have many broader applications. One thing I was confused about in my last post seems to be resolved by Jeremy’s sketch above — if I understand it correctly, one first (almost) pairs continuous measures, and only then approximates them by discrete integral measures (with a little bit of combinatorics at the end). And one really does need exponential mixing rather than just mixing.

Incidentally, apropos the matching argument, there are some interesting and well-known variations where things go haywire. For example, papers by Burago-Kleiner and (Curt) McMullen show that there are examples of separated nets in Euclidean space which are not bilipschitz to a lattice (though, interestingly, Curt shows that they are Holder equivalent). No such examples exist in hyperbolic space, because of — nonamenability and Hall’s marriage theorem! Roughly, when trying to match up points in two nets in hyperbolic space, one doesn’t need to look very far because the number of options grows exponentially. This is one reason why Kahn-Markovic need to control the matchings of their measures carefully, because it must be done on a very small scale (where the exponential growth does not kick in).

I thought I would also mention that in case my previous comments lead one to believe otherwise, exponential mixing of the geodesic flow on a hyperbolic manifold is somewhat delicate. Exponential mixing under a flow $g_t$ on a space $X$ preserving a probability measure $\mu$ means that for all (sufficiently nice) functions $f$ and $h$ on $X$, the correlations $\rho(h,f,t):= \int_X h(x)f(g_tx) d\mu - \int_X h(x) d\mu \int_X f(x) d\mu$ are bounded in absolute value by an expression of the form $C_1e^{-tC_2}$ for suitable constants $C_1,C_2$ (which might depend on the analytic quality of $f$ and $h$). For example, one takes $X$ to be the unit tangent bundle of a hyperbolic manifold, and $g_t$ the geodesic flow (i.e. the flow which pushes vectors along the geodesics they are tangent to, at constant speed). Exponential mixing should be contrasted with the much slower mixing of the horocycle flow on a hyperbolic surface, for which the correlation is bounded by an expression like $C_1(\log t)^{C_2}t^{-1}$. The geodesic flow on a hyperbolic manifold is an example of what is called an Anosov flow; i.e. the tangent bundle $TM$ splits equivariantly under the flow into three subbundles $E^0, E^s, E^u$ where $E^0$ is $1$-dimensional and tangent to the flow, $E^s$ is contracted uniformly exponentially by the flow, and $E^u$ is expanded uniformly exponentially by the flow. The best one knows for (certain) Anosov flows (by Chernov) is that the flow is stretched exponentially mixing, i.e. with an estimate of the form $C_1e^{-\sqrt{t}C_2}$. One knows exponential mixing for the geodesic flow on variable negative curvature surfaces by Dolgopyat, and on certain locally symmetric spaces, using representation theory. See Pollicott’s lecture notes here for more details. I don’t know if exponential mixing for geodesic flows is known on manifolds of variable negative curvature in high dimensions. Also I’d appreciate it if any reader who knows some ergodic theory can confirm/deny/clarify this paragraph . . .

(Update 8/12): Jeremy tells me that he and Vladimir only need “sufficiently high degree polynomial” mixing, so perhaps there is a decent chance the methods can be extended to variable negative curvature.

(Update 10/29): The paper is now available from the arXiv.

I just learned from Jesse Johnson’s blog that Vlad Markovic and Jeremy Kahn have announced a proof of the surface subgroup conjecture, that every complete hyperbolic $3$-manifold $M$ contains a closed $\pi_1$-injective surface. Equivalently, $\pi_1(M)$ contains a closed surface subgroup. Apparently, Jeremy made the announcement at an FRG conference in Utah. This answers a long-standing question in $3$-manifold topology, which is a variation on some problems originally posed by Waldhausen. If one further knew that hyperbolic $3$-manifold groups were LERF, one would be able to deduce that all hyperbolic $3$-manifolds are virtually Haken, and (by a recent theorem of Agol), virtually fibered. Dani Wise (and others) have programs to show that hyperbolic $3$-manifold groups are LERF; if successful, this would therefore resolve some of the most important outstanding problems in $3$-manifold topology (in fact, I would say: the most important outstanding problems, by a substantial margin).

In fact, the argument appears to work for hyperbolic manifolds of every dimension $\ge 3$, and possibly more generally still. Details on the argument of Markovic-Kahn are scarce (Vlad informs me that they expect to have a preprint in a few weeks) but the sketch of the argument presented by Kahn is compelling. Roughly speaking, the argument (as summarized by Ian Agol in a comment at Jesse’s blog) takes the following form:

1. Given $M$, for a sufficiently big constant $R$, one can find “many” immersed, almost totally-geodesic pairs of pants (i.e. thrice-punctured spheres) with geodesic boundary components (i.e. “cuffs”) of length very close to $2R$. In fact, one can further insist that the complex length of the boundary geodesic is very close to $2R$ (i.e. holonomy transport around this geodesic does not rotate the normal bundle very much).
2. Conversely, given any geodesic of complex length very close to $2R$, one can find many such pairs of pants that it bounds, and moreover one can find them so that the normal to the geodesic pointing in to the surface is prescribed.
3. If one takes a sufficiently big collection of such geodesic pairs of pants, one has enough of them in oppositely-aligned pairs along each boundary component, that they can be matched up (by some version of Hall’s marriage theorem), and furthermore, matched up with a definite prescribed “twist” along the boundary components
4. One checks that the resulting (closed) surface is sufficiently close to totally geodesic that the ambient negative curvature certifies it is $\pi_1$-injective

Many aspects of this argument have a lot in common with some previous attempts on the surface subgroup conjecture, including one recent approach by Bowen (note: Bowen’s approach is known to have some fatal difficulties; the “twist” in 3. above specifically addresses some of them). All of these points deserve some comments.

First, where do the pairs of pants come from? If $P$ is a totally geodesic pair of pants with boundary components of length close to $2R$, the pants $P$ retract onto a geodesic spine, i.e. an immersed totally geodesic theta graph, whose edges all have length close to $2R$, and which meet at angles very close to $120$ degrees. One can cut this spine up into two pieces, which are obtained by exponentiating the edges of an infinitesimal (almost)-planar tripod for length $R$.

Given a tripod $T$ in some plane in the tangent space at some point of $M$, one can exponentiate the edges for length $R$ to construct such a half-spine; if $T$ and $T'$ are a pair of tripods for which the exponentiated endpoints nearly match up, with almost opposite tangent vectors, then the resulting half-spines can be glued up to make a spine, and thickened to make a pair of pants. One key idea is to use the exponential mixing property of the geodesic flow on a hyperbolic manifold, e.g. as proved by Pollicott. Given some tolerance $\epsilon$, once $R$ is sufficiently large, the mixing result shows that the set of such pairs of tripods for which such a matching occurs have a definite density in the space of all pairs (and in fact, are more and more equidistributed in this space, in probability). In fact, one may even insist that two of the pairs of prongs join up to make some specific closed geodesic of length almost $2R$, and vary the pair of third prongs a very small amount so that they glue up. This takes care of the first two points; this seems quite uncontroversial (exponential mixing comes in, I suspect, to know that one doesn’t need to wiggle the pair of third prongs much, having paired the first two pairs).

The matching (i.e. the gluing up of opposite pant cuffs) apparently is done by some variant of Hall’s marriage theorem. One needs to know (I think) that for any finite set of cuffs to be glued, the set of other cuffs that they could potentially be glued to is at least as big in cardinality. This probably needs some thought, but it is plausibly true: given a cuff, it can be glued to any cuff which is almost oppositely aligned to it, and since there is some tolerance in the angle of gluing — this is where dimension at least $3$ is necessary — and moreover, since oriented cuffs are almost equidistributed, one can always find “more” cuffs that are opposite, up to a bit of tolerance, to any given subset of cuffs (of course, more details are necessary here). There is an extra wrinkle to the argument, which is that the gluing must be done with a “twist” of a definite amount, so that cuffs are not glued up in such a way that the perpendicular geodesic arcs joining pairs of cuffs match up.

(Update 8/8: I think there must necessarily be more details to the matching argument, as very loosely described above. There are at least two additional issues that must be dealt with in order to perform a matching: a parity issue (since each pants has an odd number of cuffs) and a homology issue (if the argument relativizes, so that one fixes some collection of cuffs in advance and glues up everything else, one concludes a posteriori that the union of the unglued cuffs is homologically inessential). Probably the parity issue (and more subtle divisibility issues) can be solved by gluing with real-valued weights, then approximating a real solution by a rational solution, and multiplying through to clear denominators. Maybe the homology issue does not arise, if in fact the argument doesn’t relativize.) Both these issues suggest that one does not specify in advance a collection of pants to be glued up, but rather wants to glue up a definite number of pants from some subset.)

This issue of a twist is important for the 4th point, which is perhaps the most delicate. In order to know that the resulting surface is $\pi_1$-injective, one must use geometry. A closed (immersed) surface in a hyperbolic manifold which is (locally) very close to being totally geodesic is $\pi_1$-injective. One way to see this is to observe that a geodesic loop in the surface is almost geodesic in the manifold; the ambient negative curvature means that the geodesic can be shrunk (by the negative of the gradient of length in the space of loops) to become geodesic in the ambient manifold; if it is close to being geodesic at the start, it very quickly becomes totally geodesic, without getting much shorter. Any closed geodesic in a hyperbolic manifold is essential.

If one builds a surface by gluing up almost totally geodesic pieces in such a way that there is almost no angle along the gluing, the resulting surface is almost geodesic, and therefore injective. However, one must be very careful to control the geometry of the pieces that are glued, and this is hard to do if the injectivity radius is very small. A geodesic pair of pants has area $2\pi$ no matter how long its boundary components are. So if the boundary components have length $2R$, then at the points where they are thinnest, they are only $e^{-R}$ across. If cuffs are glued where the pants are thinnest, even if the gluing angle is very small, the surfaces themselves might twist through a big angle in a very short time. So one needs to make sure that the thinnest part of one pants are glued up to a thicker part of the next, which is glued to a thicker part of the next . . . and so on. This is the point of introducing the twist before gluing: the twists accumulate, and before one has glued $R$ pieces together, one has entered the thick part of some pants, where the injectivity radius is bounded below by some universal constant.

Anyway, this seems like a really spectacular development, with an excellent chance of working out. Some of the ingredients — e.g. the exponential mixing of the geodesic flow — work just as well in variable negative curvature. In fact, some version of it should work for arbitrary hyperbolic groups (using Mineyev’s flow space). Without knowing more details of the argument, one can’t say how delicate the last part of the argument is, and how far it generalizes (but readers are invited to speculate . . .)

In a previous post, I discussed some methods for showing that a given group contains a (nonabelian) free subgroup. The methods were analytic and/or dynamical, and phrased in terms of the existence (or nonexistence) of certain functions on $G$ or on spaces derived from $G$, or in terms of actions of $G$ on certain spaces. Dually, one can try to find a free group in $G$ by finding a homomorphism $\rho: F \to G$ and looking for circumstances under which $\rho$ is injective.

For concreteness, let $G = \pi_1(X)$ for some (given) space $X$. If $F$ is a free group, a representation $\rho:F \to G$ up to conjugation determines a homotopy class of map $f: S \to X$ where $S$ is a $K(F,1)$. The most natural $K(F,1)$‘s to consider are graphs and surfaces (with boundary). It is generally not easy to tell whether a map of a graph or a surface to a topological space is $\pi_1$-injective at the topological level, but might be easier if one can use some geometry.

Example: Let $X$ be a complete Riemannian manifold with sectional curvature bounded above by some negative constant $K < 0$. Convexity of the distance function in a negatively curved space means that given any map of a graph $f:\Gamma \to X$ one can flow $f$ by the negative gradient of total length until it undergoes some topology change (e.g. some edge shrinks to zero length) or it (asymptotically) achieves a local minimum (the adjective “asymptotically” here just means that the flow takes infinite time to reach the minimum, because the size of the gradient is small when the map is almost minimum; there are no analytic difficulties to overcome when taking the limit). A typical topological change might be some loop shrinking to a point, thereby certifying that a free summand of $\pi_1(\Gamma)$ mapped trivially to $G$ and should have been discarded. Technically, one probably wants to choose $\Gamma$ to be a trivalent graph, and when some interior edge collapses (so that four points come together) to let the $4$-valent vertex resolve itself into a pair of $3$-valent vertices in whichever of the three combinatorial possibilities is locally most efficient. The limiting graph, if nonempty, will be trivalent, with geodesic edges, and vertices at which the three edges are all (tangentially) coplanar and meet at angles of $2\pi/3$. Such a graph can be certified as $\pi_1$-injective provided the edges are sufficiently long (depending on the curvature $K$). After rescaling the metric on $X$ so that the supremum of the curvatures is $-1$, a trivalent geodesic graph with angles $2\pi/3$ at the vertices and edges at least $2\tanh^{-1}(1/2) = 1.0986\cdots$ is $\pi_1$-injective. To see this, lift to maps between universal covers, i.e. consider an equivariant map from a tree $\widetilde{\Gamma}$ to $\widetilde{X}$. Let $\ell$ be an embedded arc in $\widetilde{\Gamma}$, and consider the image in $\widetilde{X}$. Using Toponogov’s theorem, one can compare with a piecewise isometric map from $\ell$ to $\mathbb{H}^n$. The worst case is when all the edges are contained in a single $\mathbb{H}^2$, and all corners “bend” the same way. Providing the image does not bend as much as a horocircle, the endpoints of the image of $\ell$ stay far away in $\mathbb{H}^2$. An infinite sided convex polygon in $\mathbb{H}^2$ with all edges of length $2\tanh^{-1}(1/2)$ and all angles $2\pi/3$ osculates a horocycle, so we are done.

Remark: The fundamental group of a negatively curved manifold is word-hyperbolic, and therefore contains many nonabelian free groups, which may be certified by pingpong applied to the action of the group on its Gromov boundary. The point of the previous example is therefore to certify that a certain subgroup is free in terms of local geometric data, rather than global dynamical data (so to speak). Incidentally, I would not swear to the correctness of the constants above.

Example: A given free group is the fundamental group of a surface with boundary in many different ways (this difference is one of the reasons that a group like $\text{Out}(F_n)$ is so much more complicated than the mapping class group of a surface). Pick a realization $F = \pi_1(S)$. Then a homomorphism $\rho:F \to G$ up to conjugacy determines a homotopy class of map from $S$ to $X$ as above. If $X$ is negatively curved as before, each boundary loop is homotopic to a unique geodesic, and we may try to find a “good” map $f:S \to X$ with boundary on these geodesics. There are many possible classes of good maps to consider:

1. Fix a conformal structure on $S$ and pick a harmonic map in the homotopy class of $f$. Such a map exists since the target is nonpositively curved, by the famous theorem of Eells-Sampson. The image is real analytic if $X$ is, and is at least as negatively curved as the target, and therefore there is an a priori upper bound on the intrinsic curvature of the image; if the supremum of the curvature on $X$ is normalized to be $-1$, then the image surface is $\text{CAT}(-1)$, which just means that pointwise it is at least as negatively curved as hyperbolic space. By Gauss-Bonnet, one obtains an a priori bound on the area of the image of $S$ in terms of the Euler characteristic (which just depends on the rank of $F$). On the other hand, this map depends on a choice of marked conformal structure on $S$, and the space of such structures is noncompact.
2. Vary over all conformal structures on $S$ and choose a harmonic map of least energy (if one exists) or find a sequence of maps that undergo a “neck pinch” as a sequence of conformal structures on $S$ degenerates. Such a neck pinch exhibits a simple curve in $S$ that is essential in $S$ but whose image is inessential in $X$; such a curve can be compressed, and the topology of $S$ simplified. Since each compression increases $\chi$, after finitely many steps the process terminates, and one obtains the desired map. This is Schoen-Yau‘s method to construct a stable minimal surface representative of $S$. When the target is $3$-dimensional, the surface may be assumed to be unbranched, by a trick due to Osserman.
3. Following Thurston, pick an ideal triangulation of $S$ (i.e. a geodesic lamination of $S$ whose complementary regions are all ideal triangles); since $S$ has boundary, we may choose such a lamination by first picking a triangulation (in the ordinary sense) with all vertices on $\partial S$ and then “spinning” the vertices to infinity. Unless $\rho$ factors through a cyclic group, there is some choice of lamination so that the image of $f$ can be straightened along the lamination, and then the image spanned with $CAT(-1)$ ideal triangles to produce a pleated surface in $X$ representing $f$ (note: if $X$ has constant negative curvature, these ideal triangles can be taken to be totally geodesic). The space of pleated surfaces in fixed (closed) $X$ of given genus is compact, so this is a reasonable class of maps to work with.
4. If $G$ is merely a hyperbolic group, one can still construct pleated surfaces, not quite in $X$, but equivariantly in Mineyev’s flow space associated to $\widetilde{X}$. Here we are not really thinking of the triangles themselves, but the geodesic laminations they bound (which carry the same information).
5. If $X$ is complete and $3$-dimensional but noncompact, the space of pleated surfaces of given genus is generally not compact, and it is not always easy to find a pleated surface where you want it. This can sometimes be remedied by shrinkwrapping; one looks for a minimal/pleated/harmonic surface subject to the constraint that it cannot pass through some prescribed set of geodesics in $X$ (which act as “barriers” or “obstacles”, and force the resulting surface to end up roughly where one wants it to).

Anyway, one way or another, one can usually find a map of a surface, or a space of maps of surfaces, representing a given homomorphism, with some kind of a priori control of the geometry. Usually, this control is not enough to certify that a given map is $\pi_1$-injective, but sometimes it might be. For instance, a totally geodesic (immersed) surface in a complete manifold of constant negative curvature is always $\pi_1$-injective, and any surface whose extrinsic curvature is small enough will also be $\pi_1$-injective.

Geometric methods to certify injectivity of free or surface groups are very useful and flexible, as far as they go. Unfortunately, I know of very few topological methods to certify injectivity. By far the most important exception is the following:

Example: In $3$-dimensions, one should look for properly embedded surfaces. If $M$ is a $3$-manifold (possibly with boundary), and $S$ is a two-sided properly embedded surface, the famous Dehn’s Lemma (proved by Papakyriakopoulos) implies that either $S$ is $\pi_1$-injective, or there is an embedded essential loop in $S$ that bounds an embedded disk in $M$ on one side of $S$. Such a loop may be compressed (i.e. $S$ may be cut open along the loop, and two copies of the compressing disk sewn in) preserving the property of embeddedness, but increasing $\chi$. After finitely many steps, either $S$ compresses away entirely, or one obtains a $\pi_1$-injective surface. One way to ensure that $S$ does not compress away entirely is to start with a surface that is essential in (relative) homology; another way is to look for a surface dual to an action (of $\pi_1(M)$) on a tree. In the latter case, one can often construct quite different free subgroups in $\pi_1(M)$ by pingpong on the ends of the tree. Note by the way that this method produces closed surface subgroups as well as free subgroups. Note too that two-sidedness is essential to apply Dehn’s Lemma.

Remark: Modern $3$-manifold topologists are sometimes unreasonably indifferent to the power of Dehn’s Lemma (probably because this tool has been incorporated so fully into their subconscious?); it is worth reading Ralph Fox’s review of Papakyriakopoulos’s paper (linked above). Of this paper, he writes:

. . . it has already led to renewed attack on the problem of classifying the 3-dimensional manifolds; significant results have been and are being obtained. A complete solution has suddenly become a definite possibility.

Remember this was written more than 50 years ago — before the geometrization conjecture, before the JSJ decomposition, before the Scott core theorem, before Haken manifolds. The only reasonable reaction to this is: !!!

Example: The construction of injective surfaces by Dehn’s Lemma may be abstracted in the following way. Given a target space $X$, and a class of maps $\mathcal{F}$ of surfaces into $X$ (in some category; e.g. homotopy classes of maps, pleated surfaces, $\text{CAT}(-1)$ surfaces, etc.) suppose one can find a complexity $c:\mathcal{F} \to \mathcal{O}$ with values in some ordered set, such that if $f \in \mathcal{F}$ is not injective, one can find $f' \in \mathcal{F}$ of smaller complexity. Then if $\mathcal{O}$ is well-ordered, an injective surface may be found. If $\mathcal{O}$ is not well-ordered, one may ask at least that $c$ is upper semi-continuous on $\mathcal{F}$, and hope to extend it upper semi-continuously to some suitable compactification of $\mathcal{F}$. Even if $\mathcal{O}$ is not well-ordered, one can at least certify that a map is injective, by showing that it minimizes $c$. Here are some potential examples (none of them entirely satisfactory).

1. Given a (homologically trivial) homotopy class of loop $\gamma$ in $X$, one can look at all maps of orientable surfaces $S$ to $X$ with boundary factoring through $\gamma$. For such a surface, let $n(S)$ denote the degree with which the (possibly multiple) boundary (components) of $S$ wrap homologically around $\gamma$, and let $-\chi^-(S)$ denote the sum of Euler characteristics of non-disk and non-sphere components of $S$. For each surface $S$, one considers the quantity $-\chi^-(S)/2n(S)$ (the factor of $2$ can be ignored if desired). The important feature of this quantity is that it does not change if $S$ is replaced by a finite cover. If $\pi_1(S)$ is not injective, let $\alpha$ be an essential loop on $S$ whose image in $X$ is inessential. Peter Scott showed that any essential loop on a surface lifts to an embedded loop in some finite cover. Hence, after passing to such a cover, $\alpha$ may be compressed, and the resulting surface $S'$ satisfies $-\chi^-(S')/2n(S') < -\chi^-(S)/2n(S)$. In other words, a global minimizer of this quantity is injective. Such a surface is called extremal. The problem is that extremal surfaces do not always exist; but this construction motivates one to look for them.
2. Given a $\text{CAT}(-1)$ surface $S$ with geodesic boundary in $X$, one can retract $S$ to a geodesic spine, and encode the surface by the resulting fatgraph, with edges labelled by homotopy classes in $X$. Since Euler characteristic is local, one does not really care precisely how the pieces of the fatgraph are assembled, but only how many pieces of what kinds are needed for a given boundary. So if only finitely many such pieces appear in some infinite family of surfaces, one can in fact construct an extremal surface as above, which is necessarily injective (more technically, one reduces the computation of Euler characteristic to a linear programming problem, finds a rational extremal solution (which corresponds to a weighted sum of pieces of fatgraph), and glues together the pieces to construct the extremal surface; one situation in which this scheme can be made to work is explained in this paper of mine). Edges can be subdivided into a finite number of possibilities, so one just needs to ensure finiteness of the number of vertex types. One condition that ensures finiteness of vertex types is the existence of a uniform constant $C>0$ so that for each surface $S$ in the given family, and for each point $p \in S$, there is an estimate $\text{dist}(p,\partial S) \le C$. If this condition is violated, one finds pairs $p_i,S_i$ which converge in the geometric topology to a point in a complete (i.e. without boundary, but probably noncompact) surface.
3. Given $S \to X$, either compress an embedded essential loop, or realize $S$ by a least area surface. If $S$ is not injective, pass to a cover, compress a loop, and realize the result by a least area surface. Repeat this process. One obtains in this way a sequence of least area surfaces in $X$ (typically of bigger and bigger genus) and there is no reason to expect the process to terminate. If $X$ is a $3$-manifold, the curvature of a least area surface admits two-sided curvature bounds away from the boundary, by a theorem of Schoen (near the boundary, the negative curvature might blow up, but only in controlled ways — e.g. after rescaling about a sequence of points with the most negative curvature, one may obtain in the limit a helicoid). Away from the boundary, the family of surfaces one obtains vary precompactly in the $C^\infty$ topology, and one may obtain a complete locally least area lamination $\Lambda$ in the limit. If $\pi_1(\Lambda)$ is not injective, one can continue to pass to covers (applying a version of Scott’s theorem for infinite surfaces) and compress, and by transfinite induction, eventually arrive at a locally least area lamination with injective $\pi_1$. Of course, such a limit might well be a lamination by planes. However, the lamination one obtains is not completely arbitrary: since it is a limit of limits of . . . compact surfaces, one can choose a limit that admits a nontrivial invariant transverse measure (one must be careful here, since the lamination will typically have boundary). Or, as in bullet 2. above, one may insist that this limit lamination is complete (i.e. without boundary).

It is more tricky to find a limit lamination as in 3. without boundary and admitting an invariant transverse measure; in any case, this motivates the following:

Question: Is there a closed hyperbolic $3$-manifold $M$ which admits a locally least area transversely measured complete immersed lamination $\Lambda$, all of whose leaves are disks? (note that the answer is negative if one asks for the lamination to be embedded (there are several easy proofs of this fact)).

Secretly, the function that assigns $\inf_S -\chi^-(S)/2n(S)$ to a homologically trivial loop $\gamma$ is the stable commutator length of the conjugacy class in $\pi_1(X)$ represented by $\gamma$. Extremal surfaces can sometimes be certified by constructing certain functions on $\pi_1(X)$ called homogeneous quasimorphisms, but a discussion of such functions will have to wait for another post.

As an experiment, I plan to spend the next five weeks documenting my current research on this blog. This research comprises several related projects, but most are concerned in one way or another with the general program of studying the geometry of a space by probing it with surfaces. Since I am nominally a topologist, these surfaces are real $2$-manifolds, and I am usually interested in working in the homotopy category (or some rational “quotient” of it). I am especially concerned with surfaces with boundary, and even (occasionally) with corners.

Since it is good to have a “big question” lurking somewhere in the background (for the purposes of motivation and advertising, if nothing else), I should admit from the start that I am interested in Gromov’s well-known question about surface subgroups, which asks:

Question (Gromov): Does every one-ended word-hyperbolic group contain a closed hyperbolic surface subgroup?

I don’t have strong feelings about whether the answer to this question is “yes” or “no”, but I do think the question can be sharpened usefully in many ways, and it is my intention to do so. Gromov’s question is certainly inspired by questions such as Waldhausen’s conjecture and the virtual fibration conjecture in $3$-manifold topology, but it is hard to imagine that a proof of one of these conjectures would shed much light on Gromov’s question in general. At least one essential tool in $3$-manifold topology — namely Dehn’s lemma — has no meaningful analogue in geometric group theory, and I think it is important to try to imagine different methods of constructing surface groups from “first principles”.

Another long-term project that informs much of my current research is the problem of understanding stable commutator length in free groups. The interested reader can learn something about this from my monograph (which can be downloaded from this page). I hope to explain why this is a fundamental and interesting problem, with rich structure and many potential applications.