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A few weeks ago, Ian Agol, Vlad Markovic, Ursula Hamenstadt and I organized a “hot topics” workshop at MSRI with the title Surface subgroups and cube complexes. The conference was pretty well attended, and (I believe) was a big success; the organizers clearly deserve a great deal of credit. The talks were excellent, and touched on a wide range of subjects, and to those of us who are mid-career or older it was a bit shocking to see how quickly the landscape of low-dimensional geometry/topology and geometric group theory has been transformed by the recent breakthrough work of (Kahn-Markovic-Haglund-Wise-Groves-Manning-etc.-) Agol. Incidentally, when I first started as a graduate student, I had a vague sense that I had somehow “missed the boat” — all the exciting developments in geometry due to Thurston, Sullivan, Gromov, Freedman, Donaldson, Eliashberg etc. had taken place 10-20 years earlier, and the subject now seemed to be a matter of fleshing out the consequences of these big breakthroughs. 20 years and several revolutions later, I no longer feel this way. (Another slightly shocking aspect of the workshop was for me to realize that I am older or about as old as 75% of the speakers . . .)

The rationale for the workshop (which I had some hand in drafting, and therefore feel comfortable quoting here) was the following:

Recently there has been substantial progress in our understanding of the related questions of which hyperbolic groups are cubulated on the one hand, and which contain a surface subgroup on the other. The most spectacular combination of these two ideas has been in 3-manifold topology, which has seen the resolution of many long-standing conjectures. In turn, the resolution of these conjectures has led to a new point of view in geometric group theory, and the introduction of powerful new tools and structures. The goal of this conference will be to explore the further potential of these new tools and perspectives, and to encourage communication between researchers working in various related fields.

I have blogged a bit about cubulated groups and surface subgroups previously, and I even began this blog (almost 4 years ago now) initially with the idea of chronicling my efforts to attack Gromov’s surface subgroup question. This question asks the following:

Gromov’s Surface Subgroup Question: Does every one-ended hyperbolic group contain a subgroup which is isomorphic to the fundamental group of a closed surface of genus at least 2?

The restriction to one-ended groups is just meant to rule out silly examples, like finite or virtually cyclic groups (i.e. “elementary” hyperbolic groups), or free products of simpler hyperbolic groups. Asking for the genus of the closed surface to be at least 2 rules out the sphere (whose fundamental group is trivial) and the torus (whose fundamental group $\mathbb{Z}^2$ cannot be a subgroup of a hyperbolic group). It is the purpose of this blog post to say that Alden Walker and I have managed to show that Gromov’s question has a positive answer for “most” hyperbolic groups; more precisely, we show that a random group (in the sense of Gromov) contains a surface subgroup (in fact, many surface subgroups) with probability going to 1 as a certain natural parameter (the “length” $n$ of the random relators) goes to infinity. (update April 8: the preprint is available from the arXiv here.)

Today Jason Manning gave a talk on a vital ingredient in the proof of Agol’s theorem, which is a result in geometric group theory. The theorem is a joint project of Agol-Groves-Manning, and generalizes some earlier work they did a few years ago. Jason referred to the main theorem during his talk as the “Goal Theorem” (I guess it was the goal of his lecture), but I’m going to call it the Weak Separation Theorem, since that is a somewhat more descriptive name. The statement of the theorem is as follows.

Weak Separation Theorem (Agol-Groves-Manning): Let G be a hyperbolic group, let H be a subgroup of G which is quasiconvex, and isomorphic to the fundamental group of a virtually special NPC cube complex, and let g be an element of G which is not contained in H. Then there is a surjection $\phi:G \to \bar{G}$ so that

1. $\bar{G}$ is hyperbolic;
2. $\phi(H)$ is finite; and
3. $\phi(g)$ is not contained in $\phi(H)$.

In the remainder of this post I will try to explain the proof of this theorem, to the extent that I understand it. Basically, this amounts to my summarizing Manning’s talk (or the part of it that I managed to get down in my notes); again, any errors, foolishness, silly blog post titles etc. are due to me.

Last Friday, Henry Wilton gave a talk at Caltech about his recent joint work with Sang-hyun Kim on polygonal words in free groups. Their work is motivated by the following well-known question of Gromov:

Question(Gromov): Let $G$ be a one-ended word-hyperbolic group. Does $G$ contain a subgroup isomorphic to the fundamental group of a closed hyperbolic surface?

Let me briefly say what “one-ended” and “word-hyperbolic” mean.

A group is said to be word-hyperbolic if it acts properly and cocompactly by isometries on a proper $\delta$-hyperbolic path metric space — i.e. a path metric space in which there is a constant $\delta$ so that geodesic triangles in the metric space have the property that each side of the triangle is contained in the $\delta$-neighborhood of the union of the other two sides (colloquially, triangles are thin). This condition distills the essence of negative curvature in the large, and was shown by Gromov to be equivalent to several other conditions (eg. that the group satisfies a linear isoperimetric inequality; that every ultralimit of the group is an $\mathbb{R}$-tree). Free groups are hyperbolic; fundamental groups of closed manifolds with negative sectional curvature (eg surfaces with negative Euler characteristic) are word-hyperbolic; “random” groups are hyperbolic — and so on. In fact, it is an open question whether a group $G$ that admits a finite $K(G,1)$ is word hyperbolic if and only if it does not contain a copy of a Baumslag-Solitar group $BS(m,n):=\langle x,y \; | \; x^{-1}y^{m}x = y^n \rangle$ for $m,n \ne 0$ (note that the group $\mathbb{Z}\oplus \mathbb{Z}$ is the special case $m=n=1$); in any case, this is a very good heuristic for identifying the word-hyperbolic groups one typically meets in examples.

If $G$ is a finitely generated group, the ends of $G$ really means the ends (as defined by Freudenthal) of the Cayley graph of $G$ with respect to some finite generating set. Given a proper topological space $X$, the set of compact subsets of $X$ gives rise to an inverse system of inclusions, where $X-K'$ includes into $X-K$ whenever $K$ is a subset of $K'$. This inverse system defines an inverse system of maps of discrete spaces $\pi_0(X-K') \to \pi_0(X-K)$, and the inverse limit of this system is a compact, totally disconnected space $\mathcal{E}(X)$, called the space of ends of $X$. A proper topological space is canonically compactified by its set of ends; in fact, the compactification $X \cup \mathcal{E}(X)$ is the “biggest” compactification of $X$ by a totally disconnected space, in the sense that for any other compactification $X \subset Y$ where $Y-X$ is zero dimensional, there is a continuous map $X \cup \mathcal{E}(X) \to Y$ which is the identity on $X$.

For a word-hyperbolic group $G$, the Cayley graph can be compactified by adding the ideal boundary $\partial_\infty G$, but this is typically not totally disconnected. In this case, the ends of $G$ can be recovered as the components of $\partial_\infty G$.

A group $G$ acts on its own ends $\mathcal{E}(G)$. An elementary argument shows that the cardinality of $\mathcal{E}(G)$ is one of $0,1,2,\infty$ (if a compact set $V$ disconnects $e_1,e_2,e_3$ then infinitely many translates of $V$ converging to $e_1$ separate $e_3$ from infinitely many other ends accumulating on $e_1$). A group has no ends if and only if it is finite. Stallings famously showed that a (finitely generated) group has at least $2$ ends if and only if it admits a nontrivial description as an HNN extension or amalgamated free product over a finite group. One version of the argument proceeds more or less as follows, at least when $G$ is finitely presented. Let $M$ be an $n$-dimensional Riemannian manifold with fundamental group $G$, and let $\tilde{M}$ denote the universal cover. We can identify the ends of $G$ with the ends of $\tilde{M}$. Let $V$ be a least ($n-1$-dimensional) area hypersurface in $\tilde{M}$ amongst all hypersurfaces that separate some end from some other (here the hypothesis that $G$ has at least two ends is used). Then every translate of $V$ by an element of $G$ is either equal to $V$ or disjoint from it, or else one could use the Meeks-Yau “roundoff trick” to find a new $V'$ with strictly lower area than $V$. The translates of $V$ decompose $\tilde{M}$ into pieces, and one can build a tree $T$ whose vertices correspond to to components of $\tilde{M} - G\cdot V$, and whose edges correspond to the translates $G\cdot V$. The group $G$ acts on this tree, with finite edge stabilizers (by the compactness of $V$), exhibiting $G$ either as an HNN extension or an amalgamated product over the edge stabilizers. Note that the special case $|\mathcal{E}(G)|=2$ occurs if and only if $G$ has a finite index subgroup which is isomorphic to $\mathbb{Z}$.

Free groups and virtually free groups do not contain closed surface subgroups; Gromov’s question more or less asks whether these are the only examples of word-hyperbolic groups with this property.

Kim and Wilton study Gromov’s question in a very, very concrete case, namely that case that $G$ is the double of a free group $F$ along a word $w$; i.e. $G = F *_{\langle w \rangle } F$ (hereafter denoted $D(w)$). Such groups are known to be one-ended if and only if $w$ is not contained in a proper free factor of $F$ (it is clear that this condition is necessary), and to be hyperbolic if and only if $w$ is not a proper power, by a result of Bestvina-Feighn. To see that this condition is necessary, observe that the double $\mathbb{Z} *_{p\mathbb{Z}} \mathbb{Z}$ is isomorphic to the fundamental group of a Seifert fiber space, with base space a disk with two orbifold points of order $p$; such a group contains a $\mathbb{Z}\oplus \mathbb{Z}$. One might think that such groups are too simple to give an insight into Gromov’s question. However, these groups (or perhaps the slightly larger class of graphs of free groups with cyclic edge groups) are a critical case for at least two reasons:

1. The “smaller” a group is, the less room there is inside it for a surface group; thus the “simplest” groups should have the best chance of being a counterexample to Gromov’s question.
2. If $G$ is word-hyperbolic and one-ended, one can try to find a surface subgroup by first looking for a graph of free groups $H$ in $G$, and then looking for a surface group in $H$. Since a closed surface group is itself a graph of free groups, one cannot “miss” any surface groups this way.

Not too long ago, I found an interesting construction of surface groups in certain graphs of free groups with cyclic edge groups. In fact, I showed that every nontrivial element of $H_2(G;\mathbb{Q})$ in such a group is virtually represented by a sum of surface subgroups. Such surface subgroups are obtained by finding maps of surface groups into $G$ which minimize the Gromov norm in their (projective) homology class. I think it is useful to extend Gromov’s question by making the following

Conjecture: Let $G$ be a word-hyperbolic group, and let $\alpha \in H_2(G;\mathbb{Q})$ be nonzero. Then some multiple of $\alpha$ is represented by a norm-minimizing surface (which is necessarily $\pi_1$-injective).

Note that this conjecture does not generalize to wider classes of groups. There are even examples of $\text{CAT}(0)$ groups $G$ with nonzero homology classes $\alpha \in H_2(G;\mathbb{Q})$ with positive, rational Gromov norm, for which there are no $\pi_1$-injective surfaces representing a multiple of $\alpha$ at all.

It is time to define polygonal words in free groups.

Definition: Let $F$ be free. Let $X$ be a wedge of circles whose edges are free generators for $F$. A cyclically reduced word $w$ in these generators is polygonal if there exists a van-Kampen graph $\Gamma$ on a surface $S$ such that:

1. every complementary region is a disk whose boundary is a nontrivial (possibly negative) power of $w$;
2. the (labelled) graph $\Gamma$ immerses in $X$ in a label preserving way;
3. the Euler characteristic of $S$ is strictly less than the number of disks.

The last condition rules out trivial examples; for example, the double of a single disk whose boundary is labeled by $w^n$. Notice that it is very important to allow both positive and negative powers of $w$ as boundaries of complementary regions. In fact, if $w$ is not in the commutator subgroup, then the sum of the powers over all complementary regions is necessarily zero (and if $w$ is in the commutator subgroup, then $D(w)$ has nontrivial $H_2$, so one already knows that there is a surface subgroup).

Condition 2. means that at each vertex of $\Gamma$, there is at most one oriented label corresponding to each generator of $F$ or its inverse. This is really the crucial geometric property. If $\Gamma,S$ is a van-Kampen graph as above, then a theorem of Marshall Hall implies that there is a finite cover of $X$ into which $\Gamma$ embeds (in fact, this observation underlies Stallings’s work on foldings of graphs). If we build a $2$-complex $Y$ with $\pi_1(Y)=D(w)$ by attaching two ends of a cylinder to suitable loops in two copies of $X$, then a tubular neighborhood of $\Gamma$ in $S$ (i.e. what is sometimes called a “fatgraph” ) embeds in a finite cover $\tilde{Y}$ of $Y$, and its double — a surface of strictly negative Euler characteristic — embeds as a closed surface in $\tilde{Y}$, and is therefore $\pi_1$-injective. Hence if $w$ is polygonal, $D(w)$ contains a surface subgroup.

Not every word is polygonal. Kim-Wilton discuss some interesting examples in their paper, including:

1. suppose $w$ is a cyclically reduced product of proper powers of the generators or their inverses (e.g a word like $a^3b^7a^{-2}c^{13}$ but not a word like $a^3bc^{-1}$); then $w$ is polygonal;
2. a word of the form $\prod_i a^{p_{2i-1}}(a^{p_{2i}})^b$ is polygonal if $|p_i|>1$ for each $i$;
3. the word $abab^2ab^3$ is not polygonal.

To see 3, suppose there were a van-Kampen diagram with more disks than Euler characteristic. Then there must be some vertex of valence at least $3$. Since $w$ is positive, the complementary regions must have boundaries which alternate between positive and negative powers of $w$, so the degree of the vertex must be even. On the other hand, since $\Gamma$ must immerse in a wedge of two circles, the degree of every vertex must be at most $4$, so there is consequently some vertex of degree exactly $4$. Since each $a$ is isolated, at least $2$ edges must be labelled $b$; hence exactly two. Hence exactly two edges are labelled $a$. But one of these must be incoming and one outgoing, and therefore these are adjacent, contrary to the fact that $w$ does not contain a $a^{\pm 2}$.

1 above is quite striking to me. When $w$ is in the commutator subgroup, one can consider van-Kampen diagrams as above without the injectivity property, but with the property that every power of $w$ on the boundary of a disk is positive; call such a van-Kampen diagram monotone. It turns out that monotone van-Kampen diagrams always exist when $w \in [F,F]$, and in fact that norm-minimizing surfaces representing powers of the generator of $H_2(D(w))$ are associated to certain monotone diagrams. The construction of such surfaces is an important step in the argument that stable commutator length (a kind of relative Gromov norm) is rational in free groups. In my paper scl, sails and surgery I showed that monomorphisms of free groups that send every generator to a power of that generator induce isometries of the $\text{scl}$ norm; in other words, there is a natural correspondence between certain equivalence classes of monotone surfaces for an arbitrary word in $[F,F]$ and for a word of the kind that Kim-Wilton show is polygonal (Note: Henry Wilton tells me that Brady, Forester and Martinez-Pedroza have independently shown that $D(w)$ contains a surface group for such $w$, but I have not seen their preprint (though I would be very grateful to get a copy!)).

In any case, if not every word is polygonal, all is not lost. To show that $D(w)$ contains a surface subgroup is suffices to show that $D(w')$ contains a surface subgroup, where $w$ and $w'$ differ by an automorphism of $F$. Kim-Wilton conjecture that one can always find an automorphism $\phi$ so that $\phi(w)$ is polygonal. In fact, they make the following:

Conjecture (Kim-Wilton; tiling conjecture): A word $w$ not contained in a proper free factor of shortest length (in a given generating set) in its orbit under $\text{Aut}(F)$ is polygonal.

If true, this would give a positive answer to Gromov’s question for groups of the form $D(w)$.

The purpose of this post is to discuss my recent paper with Koji Fujiwara, which will shortly appear in Ergodic Theory and Dynamical Systems, both for its own sake, and in order to motivate some open questions that I find very intriguing. The content of the paper is a mixture of ergodic theory, geometric group theory, and computer science, and was partly inspired by a paper of Jean-Claude Picaud. To state the results of the paper, I must first introduce a few definitions and some background.

Let $\Gamma$ be a finite directed graph (hereafter a digraph) with an initial vertex, and edges labeled by elements of a finite set $S$ in such a way that each vertex has at most one outgoing edge with any given label. A finite directed path in $\Gamma$ starting at the initial vertex determines a word in the alphabet $S$, by reading the labels on the edges traversed (in order). The set $L \subset S^*$ of words obtained in this way is an example of what is called a regular language, and is said to be parameterized by $\Gamma$. Note that this is not the most general kind of regular language; in particular, any language $L$ of this kind will necessarily be prefix-closed (i.e. if $w \in L$ then every prefix of $w$ is also in $L$). Note also that different digraphs might parameterize the same (prefix-closed) regular language $L$.

If $S$ is a set of generators for a group $G$, there is an obvious map $L \to G$ called the evaluation map that takes a word $w$ to the element of $G$ represented by that word.

Definition: Let $G$ be a group, and $S$ a finite generating set. A combing of $G$ is a (prefix-closed) regular language $L$ for which the evaluation map $L \to G$ is a bijection, and such that every $w \in L$ represents a geodesic in $G$.

The intuition behind this definition is that the set of words in $L$ determines a directed spanning tree in the Cayley graph $C_S(G)$ starting at $\text{id}$, and such that every directed path in the tree is a geodesic in $C_S(G)$. Note that there are other definitions of combing in the literature; for example, some authors do not require the evaluation map to be a bijection, but only a coarse bijection.

Fundamental to the theory of combings is the following Theorem, which paraphrases one of the main results of this paper:

Theorem: (Cannon) Let $G$ be a hyperbolic group, and let $S$ be a finite generating set. Choose a total order on the elements of $S$. Then the language $L$ of lexicographically first geodesics in $G$ is a combing.

The language $L$ described in this theorem is obviously geodesic and prefix-closed, and the evaluation map is bijective; the content of the theorem is that $L$ is regular, and parameterized by some finite digraph $\Gamma$. In the sequel, we restrict attention exclusively to hyperbolic groups $G$.

Given a (hyperbolic) group $G$, a generating set $S$, a combing $L$, one makes the following definition:

Definition: A function $\phi:G \to \mathbb{Z}$ is weakly combable (with respect to $S,L$) if there is a digraph $\Gamma$ parameterizing $L$ and a function $d\phi$ from the vertices of $\Gamma$ to $\mathbb{Z}$ so that for any $w \in L$, corresponding to a path $\gamma$ in $\Gamma$, there is an equality $\phi(w) = \sum_i d\phi(\gamma(i))$.

In other words, a function $\phi$ is weakly combable if it can be obtained by “integrating” a function $d\phi$ along the paths of a combing. One furthermore says that a function is combable if it changes by a bounded amount under right-multiplication by an element of $S$, and bicombable if it changes by a bounded amount under either left or right multiplication by an element of $S$. The property of being (bi-)combable does not depend on the choice of a generating set $S$ or a combing $L$.

Example: Word length (with respect to a given generating set $S$) is bicombable.

Example: Let $\phi:G \to \mathbb{Z}$ be a homomorphism. Then $\phi$ is bicombable.

Example: The Brooks counting quasimorphisms (on a free group) and the Epstein-Fujiwara counting quasimorphisms are bicombable.

Example: The sum or difference of two (bi-)combable functions is (bi-)combable.

A particularly interesting example is the following:

Example: Let $S$ be a finite set which generates $G$ as a semigroup. Let $\phi_S$ denote word length with respect to $S$, and $\phi_{S^{-1}}$ denote word length with respect to $S^{-1}$ (which also generates $G$ as a semigroup). Then the difference $\psi_S:= \phi_S - \phi_{S^{-1}}$ is a bicombable quasimorphism.

The main theorem proved in the paper concerns the statistical distribution of values of a bicombable function.

Theorem: Let $G$ be a hyperbolic group, and let $\phi$ be a bicombable function on $G$. Let $\overline{\phi}_n$ be the value of $\phi$ on a random word in $G$ of length $n$ (with respect to a certain measure $\widehat{\nu}$ depending on a choice of generating set). Then there are algebraic numbers $E$ and $\sigma$ so that as distributions, $n^{-1/2}(\overline{\phi}_n - nE)$ converges to a normal distribution with standard deviation $\sigma$.

One interesting corollary concerns the length of typical words in one generating set versus another. The first thing that every geometric group theorist learns is that if $S_1, S_2$ are two finite generating sets for a group $G$, then there is a constant $K$ so that every word of length $n$ in one generating set has length at most $nK$ and at least $n/K$ in the other generating set. If one considers an example like $\mathbb{Z}^2$, one sees that this is the best possible estimate, even statistically. However, if one restricts attention to a hyperbolic group $G$, then one can do much better for typical words:

Corollary: Let $G$ be hyperbolic, and let $S_1,S_2$ be two finite generating sets. There is an algebraic number $\lambda_{1,2}$ so that almost all words of length $n$ with respect to the $S_1$ generating set have length almost equal to $n\lambda_{1,2}$ with respect to the $S_2$ generating set, with error of size $O(\sqrt{n})$.

Let me indicate very briefly how the proof of the theorem goes.

Sketch of Proof: Let $\phi$ be bicombable, and let $d\phi$ be a function from the vertices of $\Gamma$ to $\mathbb{Z}$, where $\Gamma$ is a digraph parameterizing $L$. There is a bijection between the set of elements in $G$ of word length $n$ and the set of directed paths in $\Gamma$ of length $n$ that start at the initial vertex. So to understand the distribution of $\phi$, we need to understand the behaviour of a typical long path in $\Gamma$.

Define a component of $\Gamma$ to be a maximal subgraph with the property that there is a directed path (in the component) from any vertex to any other vertex. One can define a new digraph $C(\Gamma)$ without loops, with one vertex for each component of $\Gamma$, in an obvious way. Each component $C$ determines an adjacency matrix $M_C$, with $ij$-entry equal to $1$ if there is a directed edge from vertex $i$ to vertex $j$, and equal to $0$ otherwise. A component $C$ is big if the biggest real eigenvalue $\lambda$ of $M_C$ is at least as big as the biggest real eigenvalue of the matrices associated to every other component. A random long walk in $\Gamma$ will spend most of its time entirely in big components, so these are the only components we need to consider to understand the statistical distribution of $\phi$.

A theorem of Coornaert implies that there are no big components of $C(\Gamma)$ in series; i.e. there are no directed paths in $C(\Gamma)$ from one big component to another (one also says that the big components do not communicate). This means that a typical long walk in $\Gamma$ is entirely contained in a single big component, except for a (relatively short) path at the start and the end of the walk. So the distribution of $\phi$ gets independent contributions, one from each big component.

The contribution from an individual big component is not hard to understand: the central limit theorem for stationary Markov chains says that for elements of $G$ corresponding to paths that spend almost all their time in a given big component $C$ there is a central limit theorem  $n^{-1/2}(\overline{\phi}_n - nE_C) \to N(0,\sigma_C)$ where the mean $E_C$ and standard deviation $\sigma_C$ depend only on $C$. The problem is to show that the means and standard deviations associated to different big components are the same. Everything up to this point only depends on weak combability of $\phi$; to finish the proof one must use bicombability.

It is not hard to show that if $\gamma$ is a typical infinite walk in a component $C$, then the subpaths of $\gamma$ of length $n$ are distributed like random walks of length $n$ in $C$. What this means is that the mean and standard deviation $E_C,\sigma_C$ associated to a big component $C$ can be recovered from the distribution of $\phi$ on a single infinite “typical” path in $C$. Such an infinite path corresponds to an infinite geodesic in $G$, converging to a definite point in the Gromov boundary $\partial G$. Another theorem of Coornaert (from the same paper) says that the action of $G$ on its boundary $\partial G$ is ergodic with respect to a certain natural measure called a Patterson-Sullivan measure (see Coornaert’s paper for details). This means that there are typical infinite geodesics $\gamma,\gamma'$ associated to components $C$ and $C'$ for which some $g \in G$ takes $\gamma$ to a geodesic $g\gamma$ ending at the same point in $\partial G$ as $\gamma'$. Bicombability implies that the values of $\phi$ on $\gamma$ and $g\gamma$ differ by a bounded amount. Moreover, since $g\gamma$ and $\gamma'$ are asymptotic to the same point at infinity, combability implies that the values of $\phi$ on $g\gamma$ and $\gamma'$ also differ by a bounded amount. This is enough to deduce that $E_C = E_{C'}$ and $\sigma_C = \sigma_{C'}$, and one obtains a (global) central limit theorem for $\phi$ on $G$. qed.

This obviously raises several questions, some of which seem very hard, including:

Question 1: Let $\phi$ be an arbitrary quasimorphism on a hyperbolic group $G$ (even the case $G$ is free is interesting). Does $\phi$ satisfy a central limit theorem?

Question 2: Let $\phi$ be an arbitrary quasimorphism on a hyperbolic group $G$. Does $\phi$ satisfy a central limit theorem with respect to a random walk on $G$? (i.e. one considers the distribution of values of $\phi$ not on the set of elements of $G$ of word length $n$, but on the set of elements obtained by a random walk on $G$ of length $n$, and lets $n$ go to infinity)

All bicombable quasimorphisms satisfy an important property which is essential to our proof of the central limit theorem: they are local, which is to say, they are defined as a sum of local contributions. In the continuous world, they are the analogue of the so-called de Rham quasimorphisms on $\pi_1(M)$ where $M$ is a closed negatively curved Riemannian manifold; such quasimorphisms are defined by choosing a $1$-form $\alpha$, and defining $\phi_\alpha(g)$ to be equal to the integral $\int_{\gamma_g} \alpha$, where $\gamma_g$ is the closed oriented based geodesic in $M$ in the homotopy class of $g$. De Rham quasimorphisms, being local, also satisfy a central limit theorem.

This locality manifests itself in another way, in terms of defects. Let $\phi$ be a quasimorphism on a hyperbolic group $G$. Recall that the defect $D(\phi)$ is the supremum of $|\phi(gh) - \phi(g) -\phi(h)|$ over all pairs of elements $g,h \in G$. A quasimorphism is further said to be homogeneous if $\phi(g^n) = n\phi(g)$ for all integers $n$. If $\phi$ is an arbitrary quasimorphism, one may homogenize it by taking a limit $\psi(g) = \lim_{n \to \infty} \phi(g^n)/n$; one says that $\psi$ is the homogenization of $\phi$ in this case. Homogenization typically does not preserve defects; however, there is an inequality $D(\psi) \le 2D(\phi)$. If $\phi$ is local, one expects this inequality to be an equality. For, in a hyperbolic group, the contribution to the defect of a local quasimorphism all arises from the interaction of the suffix of (a geodesic word representing the element) $g$ with the prefix of $h$ (with notation as above). When one homogenizes, one picks up another contribution to the defect from the interaction of the prefix of $g$ with the suffix of $h$; since these two contributions are essentially independent, one expects that homogenizing a local quasimorphism should exactly double the defect. This is the case for bicombable and de Rham quasimorphisms, and can perhaps be used to define locality for a quasimorphism on an arbitrary group.

This discussion provokes the following key question:

Question 3: Let $G$ be a group, and let $\psi$ be a homogeneous quasimorphism. Is there a quasimorphism $\phi$ with homogenization $\psi$, satisfying $D(\psi) = 2D(\phi)$?

Example: The answer to question 3 is “yes” if $\psi$ is the rotation quasimorphism associated to an action of $G$ on $S^1$ by orientation-preserving homeomorphisms (this is nontrivial; see Proposition 4.70 from my monograph).

Example: Let $C$ be any homologically trivial group $1$-boundary. Then there is some extremal homogeneous quasimorphism $\psi$ for $C$ (i.e. a quasimorphism achieving equality $\text{scl}(C) = \psi(C)/2D(\psi)$ under generalized Bavard duality; see this post) for which there is $\phi$ with homogenization $\psi$ satisfying $D(\psi) = 2D(\phi)$. Consequently, if every point in the boundary of the unit ball in the $\text{scl}$ norm is contained in a unique supporting hyperplane, the answer to question 3 is “yes” for any quasimorphism on $G$.

Any quasimorphism on $G$ can be pulled back to a quasimorphism on a free group, but this does not seem to make anything easier. In particular, question 3 is completely open (as far as I know) when $G$ is a free group. An interesting test case might be the homogenization of an infinite sum of Brooks functions $\sum_w h_w$ for some infinite non-nested family of words $\lbrace w \rbrace$.

If the answer to this question is false, and one can find a homogeneous quasimorphism $\psi$ which is not the homogenization of any “local” quasimorphism, then perhaps $\psi$ does not satisfy a central limit theorem. One can try to approach this problem from the other direction:

Question 4: Given a function $f$ defined on the ball of radius $n$ in a free group $F$, one defines the defect $D(f)$ in the usual way, restricted to pairs of elements $g,h$ for which $g,h,gh$ are all of length at most $n$. Under what conditions can $f$ be extended to a function on the ball of radius $n+1$ without increasing the defect?

If one had a good procedure for building a quasimorphism “by hand” (so to speak), one could try to build a quasimorphism that failed to satisfy a central limit theorem, or perhaps find reasons why this was impossible.

More ambitious than simply showing that a group is infinite is to show that it contains an infinite subgroup of a certain kind. One of the most important kinds of subgroup to study are free groups. Hence, one is interested in the question:

Question: When does a group contain a (nonabelian) free subgroup?

Again, one can (and does) ask this question both about a specific group, and about certain classes of groups, or for a typical (in some sense) group from some given family.

Example: If $\mathcal{P}$ is a property of groups that is inherited by subgroups, then if no free group satisfies $\mathcal{P}$, no group that satisfies $\mathcal{P}$ can contain a free subgroup. An important property of this kind is amenability. A (discrete) group $G$ is amenable if it admits an invariant mean; that is, if there is a linear map $m: L^\infty(G) \to \mathbb{R}$ (i.e. a way to define the average of a bounded function over $G$) satisfying three basic properties:

1. $m(f) \ge 0$ if $f\ge 0$ (i.e. the average of a non-negative function is non-negative)
2. $m(\chi_G)=1$ where $\chi_G$ is the constant function taking the value $1$ everywhere on $G$ (i.e. the average of the constant function $1$ is normalized to be $1$)
3. $m(g\cdot f) = m(f)$ for every ${}g \in G$ and $f \in L^\infty(G)$, where $(g\cdot f)(x) = f(g^{-1}x)$ (i.e. the mean is invariant under the obvious action of $G$ on $L^\infty(G)$)

If $H$ is a subgroup of $G$, there are (many) $H$-invariant homomorphisms $j: L^\infty(H) \to L^\infty(G)$ taking non-negative functions to non-negative functions, and $\chi_H$ to $\chi_G$; for example, the (left) action of $H$ on $G$ breaks up into a collection of copies of $H$ acting on itself, right-multiplied by a collection of right coset representatives. After choosing such a choice of representatives $\lbrace g_\alpha \rbrace$, one for each coset $Hg_\alpha$, we can define $j(f)(hg_\alpha) = f(h)$. Composing with $m$ shows that every subgroup of an amenable group is amenable (this is harder to see in the “geometric” definition of amenable groups in terms of Folner sets). On the other hand, as is well-known, a nonabelian free group is not amenable. Hence, amenable groups do not contain nonabelian free subgroups.

The usual way to see that a nonabelian free group is not amenable is to observe that it contains enough disjoint “copies” of big subsets. For concreteness, let $F$ denote the free group on two generators $a,b$, and write their inverses as $A,B$. Let $W_a, W_A$ denote the set of reduced words that start with either $a$ or $A$, and let $\chi_a,\chi_A$ denote the indicator functions of $W_a,W_A$ respectively. We suppose that $F$ is amenable, and derive a contradiction. Note that $F = W_a \cup aW_A$, so $m(\chi_a) + m(\chi_A) \ge 1$. Let $V$ denote the set of reduced words that start with one of the strings $a,A,ba,bA$, and let $\chi_V$ denote the indicator function of $V$. Notice that $V$ is made of two disjoint copies of each of $W_a,W_A$. So on the one hand, $m(\chi_V) \le m(\chi_F) = 1$, but on the other hand, $m(\chi_V) = 2 (m(\chi_a)+m(\chi_A)) \ge 2$.

Conversely, the usual way to show that a group $G$ is amenable is to use the Folner condition. Suppose that $G$ is finitely generated by some subset $S$, and let $C$ denote the Cayley graph of $G$ (so that $C$ is a homogeneous locally finite graph). Suppose one can find finite subsets $U_i$ of vertices so that $|\partial U_i|/|U_i| \to 0$ (here $|U_i|$ means the number of vertices in $U_i$, and  $|\partial U_i|$ means the number of vertices in $U_i$ that share an edge with $C - U_i$). Since the “boundary” of $U_i$ is small compared to $U_i$, averaging a bounded function over $U_i$ is an “almost invariant” mean; a weak limit (in the dual space to $L^\infty(G)$) is an invariant mean. Examples of amenable groups include

1. Finite groups
2. Abelian groups
3. Unions and extensions of amenable groups
4. Groups of subexponential growth

and many others. For instance, virtually solvable groups (i.e. groups containing a solvable subgroup with finite index) are amenable.

Example: No amenable group can contain a nonabelian free subgroup. The von Neumann conjecture asked whether the converse was true. This conjecture was disproved by Olshanskii. Subsequently, Adyan showed that the infinite free Burnside groups are not amenable. These are groups $B(m,n)$ with $m\ge 2$ generators, and subject only to the relations that the $n$th power of every element is trivial. When $n$ is odd and at least $665$, these groups are infinite and nonamenable. Since they are torsion groups, they do not even contain a copy of $\mathbb{Z}$, let alone a nonabelian free group!

Example: The Burnside groups are examples of groups that obey a law; i.e. there is a word $w(x_1,x_2,\cdots,x_n)$ in finitely many free variables, such that $w(g_1,g_2,\cdots,g_n)=\text{id}$ for every choice of $g_1,\cdots,g_n \in G$. For example, an abelian group satisfies the law $x_1x_2x_1^{-1}x_2^{-1}=\text{id}$. Evidently, a group that obeys a law does not contain a nonabelian free subgroup. However, there are examples of groups which do not obey a law, but which also do not contain any nonabelian free subgroup. An example is the classical Thompson’s group $F$, which is the group of orientation-preserving piecewise-linear homeomorphisms of $[0,1]$ with finitely many breakpoints at dyadic rationals (i.e. points of the form $p/2^q$ for integers $p,q$) and with slopes integral powers of $2$. To see that this group does not obey a law, one can show (quite easily) that in fact $F$ is dense (in the $C^0$ topology) in the group $\text{Homeo}^+([0,1])$ of all orientation-preserving homeomorphisms of the interval. This latter group contains nonabelian free groups; by approximating the generators of such a group arbitrarily closely, one obtains pairs of elements in $F$ that do not satisfy any identity of length shorter than any given constant. On the other hand, a famous theorem of Brin-Squier says that $F$ does not contain any nonabelian free subgroup. In fact, the entire group $\text{PL}^+([0,1])$ does not contain any nonabelian free subgroup. A short proof of this fact can be found in my paper as a corollary of the fact that every subgroup $G$ of $\text{PL}^+([0,1])$ has vanishing stable commutator length; since stable commutator length is nonvanishing in nonabelian free groups, this shows that there are no such subgroups of $\text{PL}^+([0,1])$. (Incidentally, and complementarily, there is a very short proof that stable commutator length vanishes on any group that obeys a law; we will give this proof in a subsequent post).

Example: If $G$ surjects onto $H$, and $H$ contains a free subgroup $F$, then there is a section from $F$ to $G$ (by freeness), and therefore $G$ contains a free subgroup.

Example: The most useful way to show that $G$ contains a nonabelian free subgroup is to find a suitable action of $G$ on some space $X$. The following is known as Klein’s ping-pong lemma. Suppose one can find disjoint subsets $U^\pm$ and $V^\pm$ of $X$, and elements $g,h \in G$ so that $g(U^+ \cup V^\pm) \subset U^+$$g^{-1}(U^- \cup V^\pm) \subset U^-$, and similarly interchanging the roles of $U^\pm, V^\pm$ and $g,h$. If $w$ is a reduced word in $g^{\pm 1},h^{\pm 1}$, one can follow the trajectory of a point under the orbit of subwords of $w$ to verify that $w$ is nontrivial. The most common way to apply this in practice is when $g,h$ act on $X$ with source-sink dynamics; i.e. the element $g$ has two fixed points $u^\pm$ so that every other point converges to $u^+$ under positive powers of $g$, and to $u^-$ under negative powers of $g$. Similarly, $h$ has two fixed points $v^\pm$ with similar dynamics. If the points $u^\pm,v^\pm$ are disjoint, and $X$ is compact, one can take any small open neighborhoods $U^\pm,V^\pm$ of $u^\pm,v^\pm$, and then sufficiently large powers of $g$ and $h$ will satisfy the hypotheses of ping-pong.

Example: Every hyperbolic group $G$ acts on its Gromov boundary $\partial_\infty G$. This boundary is the set of equivalence classes of quasigeodesic rays in (the Cayley graph of) $G$, where two rays are equivalent if they are a finite Hausdorff distance apart. Non-torsion elements act on the boundary with source-sink dynamics. Consequently, every pair of non-torsion elements in a hyperbolic group either generate a virtually cyclic group, or have powers that generate a nonabelian free group.

It is striking to see how easy it is to construct nonabelian free subgroups of a hyperbolic group, and how difficult to construct closed surface subgroups. We will return to the example of hyperbolic groups in a future post.

Example: The Tits alternative says that any linear group $G$ (i.e. any subgroup of $\text{GL}(n,\mathbb{R})$ for some $n$) either contains a nonabelian free subgroup, or is virtually solvable (and therefore amenable). This can be derived from ping-pong, where $G$ is made to act on certain spaces derived from the linear action (e.g. locally symmetric spaces compactified in certain ways, and buildings associated to discrete valuations on the ring of entries of matrix elements of $G$).

Example: There is a Tits alternative for subgroups of other kinds of groups, for example mapping class groups, as shown by Ivanov and McCarthy. The mapping class group (of a surface) acts on the Thurston boundary of Teichmuller space. Every subgroup of the mapping class group either contains a nonabelian free subgroup, or is virtually abelian. Roughly speaking, either elements move points in the boundary with enough dynamics to be able to do ping-pong, or else the action is “localized” in a train-track chart, and one obtains a linear representation of the group (enough to apply the ordinary Tits alternative). Virtually solvable subgroups of mapping class groups are virtually abelian.

Example: A similar Tits alternative holds for $\text{Out}(F_n)$. This was shown by Bestvina-Feighn-Handel in these three papers (the third paper shows that solvable subgroups are virtually abelian, thus emphasizing the parallels with mapping class groups).

Example: If $G$ is a finitely generated group of homeomorphisms of $S^1$, then there is a kind of Tits alternative, first proposed by Ghys, and proved by Margulis: either $G$ preserves a probability measure on $S^1$ (which might be singular), or it contains a nonabelian free subgroup. To see this, first note that either $G$ has a finite orbit (which supports an invariant probability measure) or the action is semi-conjugate to a minimal action (one with all orbits dense). In the second case, the proof depends on understanding the centralizer of the group action: either the centralizer is infinite, in which case the group is conjugate to a group of rotations, or it is finite cyclic, and one obtains an action of $G$ on a “smaller” circle, by quotienting out by the centralizer. So one may assume the action is minimal with trivial centralizer. In this case, one shows that the action has the property that for any nonempty intervals $I,J$ in $S^1$, there is some ${}g \in G$ with $g(I) \subset J$; i.e. any interval may be put inside any other interval by some element of the group. For such an action, it is very easy to do ping-pong. Incidentally, a minor variation on this result, and with essentially this argument, was established by Thurston in the context of uniform foliations of $3$-manifolds before Ghys proposed his question.

Example: If $\rho_t$ is an (algebraic) family of representations of a (countable) free group $F$ into an algebraic group, then either some element $g \in F$ is in the kernel of every $\rho_t$, or the set of faithful representations is “generic”, i.e. the intersection of countably many open dense sets. This is because the set of representations for which a given element is in the kernel is Zariski closed, and therefore its complement is open and either empty or dense (one must add suitable hypotheses or conditions to the above to make it rigorous).

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.