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Geometric group theory is not a coherent and unified field of enquiry so much as a collection of overlapping methods, examples, and contexts. The most important examples of groups are those that arise in nature: free groups and fundamental groups of surfaces, the automorphism groups of the same, lattices, Coxeter and Artin groups, and so on; whereas the most important properties of groups are those that lend themselves to applications or can be used in certain proof templates: linearity, hyperbolicity, orderability, property (T), coherence, amenability, etc. It is natural to confront examples arising in one context with properties that arise in the other, and this is the source of a wealth of (usually very difficult) problems; e.g. do mapping class groups have property (T)? (no, by Andersen) or: is every lattice in \text{PSL}(2,\mathbb{C}) virtually orderable?

As remarked above, it is natural to formulate these questions, but not necessarily productive. Gromov, in his essay Spaces and Questions remarks that

often the mirage of naturality lures us into featureless desert with no clear perspective where the solution, even if found, does not quench our thirst for structural mathematics . . . Another approach . . . has a better chance for a successful outcome with questions following (rather than preceding) construction of new objects.

A famous question of the kind Gromov warns against is the following:

Question: Is Thompson’s group F amenable?

Recall that Thompson’s group is the group of (orientation-preserving) PL homeomorphisms of the unit interval with breakpoints at dyadic rationals (i.e. rational numbers of the form p/2^q for integers p,q) and derivatives all powers of 2. This group is a rich source of examples/counterexamples in geometric group theory: it is finitely presented (in fact FP_\infty) but “looks like” a transformation group; it contains no nonabelian free group (by Brin-Squier), but obeys no law. It is not elementary amenable (i.e. it cannot be built up from finite or abelian groups by elementary operations — subgroups, quotients, extensions, directed unions), so it is “natural” to wonder whether it is amenable at all, or whether it is one of the rare examples of nonamenable groups without nonabelian free subgroups (see this post for a discussion of amenability versus the existence of free subgroups, and von Neumann’s conjecture). This question has attracted a great deal of attention, possibly because of its long historical pedigree, rather than because of the potential applications of a positive (or negative) answer.

Recently, two papers were posted on the arXiv, promising competing resolutions of the question. In February, Azer Akhmedov posted a preprint claiming to show that the group F is not amenable. This preprint was revised, withdrawn, then revised again, and as of the end of April, Akhmedov continues to press his claim. Akhmedov’s argument depends on a new geometric criterion for nonamenability, roughly speaking, the existence of a 2-generator subgroup and a subadditive non-negative function on the group whose values grow at a definite rate on words in the subgroup whose exponents satisfy suitable parity conditions and inequalities. The non-negative function (Akhmedov calls it a “height function”) certifies the existence of a sufficiently “bushy” subset of the group to violate Folner’s criterion for amenability. Akhmedov’s paper reads like a “conventional” paper in geometric group theory, using methods from coarse geometry, careful combinatorial and counting arguments to establish the existence of a geometric object with certain large-scale properties, and an appeal to a standard geometric criterion to obtain the desired result. Akhmedov’s paper is part of a series, relating (non)amenability to certain other interesting geometric properties, some related to the so-called “traveling salesman” property, introduced earlier by Akhmedov.

On the other hand, in May, E. Shavgulidze posted a preprint claiming to show that the group F is amenable. Interestingly enough, Shavgulidze’s argument does not apply to the slightly more general class of Stein-Thompson groups in which slopes and denominators of break points can be divisible by an arbitrary (but prescribed) finite set of prime numbers. Moreover, his methods are very unlike any that one would expect to find in the typical geometric group theory paper. The argument depends on the construction, going back (in some sense) to a paper of Shavgulidze from 1978, of a measure on the space C(I) of continuous functions on the interval which is quasi-invariant under the natural action of the group of diffeomorphisms of the interval of regularity at least C^3. In more detail, let D^n denote the group of diffeomorphisms of the interval of regularity at least C^n for each n, and let C denote the Banach space of continuous functions on the interval that vanish at the origin. Define A:D^1 \to C by the formula A(f)(t) = \log(f'(t)) - \log(f'(0)). The space C can be equipped with a natural measure — the Wiener measure w_\sigma of variance \sigma, and this measure can be pulled back to D^1 by A, which is thought of as a topological space with the C^1 topology. Shavgulidze shows that the left action of D^3 on D^1 quasi-preserves this measure. Here the Wiener measure on C is the probability measure associated to Brownian motion (with given variance). A “sample” trajectory W_t from C is characterized by three properties: that it starts at the origin (i.e. W_0=0), that is it continuous almost surely (this is already implicit in the fact that the measure is supported on the space C and not some more general space), and that increments are independent, with the distribution of W_t - W_s equal to a Gaussian with mean zero and variance (t-s)\sigma. Shavgulidze’s argument depends first on an argument of Ghys-Sergiescu that shows Thompson’s group is conjugate (by a homeomorphism) to a discrete subgroup of the group of C^\infty diffeomorphisms of the interval. A bounded function f on F determines a continuous bounded function \pi_\delta(f) on D^{1+\delta} (for \delta<1/2) by a certain convolution trick, using both the group structure of F, and its discreteness in D^3. Roughly, given an element g \in D^{1+\delta}, the set of elements of F whose (group) composition with g is uniformly bounded in the C^{1+\delta} norm is finite; so the value of \pi_\delta(f) is obtained by taking a suitable average of the value of f on this finite subset of F. This reduces the problem of the amenability of F to the existence of a suitable functional on the space of bounded continuous functions on D^{1+\delta}, which is constructed via the pulled back Wiener measure as above.

There are several distinctive features of Shavgulidze’s preprint. One of the most striking is that it depends on very delicate analytic features of the Wiener measure, and the way it transforms under the action of D^3 on D^1 — a transformation law involving the Schwartzian derivative — and suggesting that certain parts of the argument could be clarified (at least from the point of view of a topologist?) by using projective geometry and Sturm-Liouville theory. Another is that the crucial analytic quality — namely differentiability of class C^{1+1/2} — is also crucial for many other natural problems in 1-dimensional analysis and geometry, from regularity estimates in the thin obstacle problem, to Navas’ work on actions of property (T) groups on the circle. At least one of the preprints by Akhmedov and Shavgulidze must be in error (in fact, a real skeptic’s skeptic such as Michael Aschbacher is not even willing to concede that much . . .) but even if wrong, it is possible that they contain things more valuable than a resolution of the question that prompted them.

Update (7/6): Azer Akhmedov sent me a construction of a (nonabelian) free subgroup of D^1 that is discrete in the C^1 topology. This is not quite enough regularity to intersect with Shavgulidze’s program, but it is interesting, and worth explaining. This is my (minor) modification of Azer’s construction (any errors are due to me):

Proposition: The group D^1 contains a discrete nonabelian free subgroup.

Sketch of Proof: First, decompose the interval [0,1] into countably many disjoint subintervals accumulating only at the endpoints. Choose a free action on two generators by doing something generic on each subinterval, in such a way that the derivative is equal to 1 at the endpoints. This can certainly be accomplished; for concreteness, choose the action so that for each subinterval I_i there is a point x_i in the interior of I_i whose stabilizer is trivial.

Second, for each pair of distinct words in the generators, choose a subinterval and modify the action there so that the derivatives of those words in that subinterval differ by at least some definite constant C at some point. In more detail: enumerate the pairs of words somehow p_1, p_2, p_3 where each p_i is a pair of words (w_{i1}, w_{i2}) in the generators, and modify the action on the subinterval I_i so the words in p_i differ by at least C in the C^1 norm on the interval I_i. Since we are modifying the generators infinitely many times, but in such a way that the support of the modification exits any compact subset of the interior, we just need to check that the modifications are C^1. Since there are only finitely many pairs of words, both of which are of bounded length (for any given bound), when i is sufficiently big, one of the words w_{i1},  w_{i2} has length at least n(i) where n(i) goes to infinity as i goes to infinity. Without loss of generality, we can order the pairs so that w_{i1} is the “long” word.

Now this is how we modify the action in I_i. Recall that the point x_i has trivial stabilizer, so the translates y_{ij} of x_i under the suffixes of w_{i1} are distinct. Take disjoint intervals about the y_{ij} and observe that each y_{ij} is taken to y_{ij+1} by one of the generators. Modify this generator inside this disjoint neighborhood so that y_{ij} is still taken to y_{ij+1}, but the derivative at that point is multiplied by 1+ C/n(i), and the derivative at nearby points is not multiplied by more than 1+C/n(i). Since the neighborhoods of the y_{ij} are disjoint, these modifications are all compatible, and the derivative of the generators does not change by more than 1+C/n(i) at any point. Since n(i) goes to infinity as i goes to infinity, we can perform such modifications for each i, and the resulting action is still C^1. But now the derivative of w_{i1} at x_i has been multiplied by 1+C, so w_{i1} and w_{i2} differ by at least C in the C^1 norm.  qed.

It is interesting to observe that this construction, while C^1, is not C^{1+\epsilon} for any \epsilon>0. For big i, we have n(i) \sim \log(i) whereas |I_i| = o(1/i). Introducing a “bump” which modifies the derivative by 1/\log(i) in a subinterval of size o(1/i) will blow up every Holder norm.

(Update 8/10): Mark Sapir has created a webpage to discuss Shavgulidze’s paper here. Also, Matt Brin has posted notes on Shavgulidze’s paper here. The notes are very nice, and go into great detail, as far as they go. Matt promises to update the notes periodically.

(Update 11/18): Matt Brin has let me know by email that a significant gap has emerged in Shavgulidze’s argument. He writes:

Lemma 5 is still unproven. It claims a property about the distributions u_n on the simplexes D_n that is needed for the second part of the paper. The main result does not need the particular distributions u_n given in the paper, but does need distributions on the D_n that satisfy the properties claimed by Lemmas 5, 6 and that cooperate with Lemma 9. Ufe Haagerup claims an argument that the u_n in the paper does not satisfy the conclusion of Lemma 5. Another distribution was said to be suggested by Shavgulidze, but at last report, it did not seem to be working out.

In light of this, it would seem to be reasonable to consider the question of whether F is amenable as wide open.

(Update 9/21/2012): Justin Moore has posted a preprint on the arXiv claiming to prove amenability of F. It is too early to suggest that there is expert consensus on the correctness of the proof, but certainly everything I have heard is promising. I have not had time to look carefully at the argument yet, but hope to get a chance to do so before too long.

(Update 10/2/2012): Justin has withdrawn his claim of a proof. A gap was found by Akhmedov.

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 nth 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).

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