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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.