You are currently browsing the tag archive for the ‘Dynamics’ tag.

I have struggled for a long time (and I continue to struggle) with the following question:

**Question: **Is the group of self-homeomorphisms of the unit disk (in the plane) that fix the boundary pointwise a left-orderable group?

Recall that a group is left-orderable if there is a total order on the elements satisfying if and only if for all . For a countable group, the property of being left orderable is equivalent to the property that the group admits a faithful action on the interval by orientation-preserving homeomorphisms; however, this equivalence is not “natural” in the sense that there is no natural way to extract an ordering from an action, or vice-versa. This formulation of the question suggests that one is trying to embed the group of homeomorphisms of the disk into the group of homeomorphisms of the interval, an unlikely proposition, made even more unlikely by the following famous theorem of Filipkiewicz:

**Theorem:** (Filipkiewicz) Let be two compact manifolds, and two non-negative integers or infinity. Suppose the connected components of the identity of and are isomorphic as abstract groups. Then and the isomorphism is induced by some diffeomorphism.

The hard(est?) part of the argument is to identify a subgroup stabilizing a point in purely algebraic terms. It is a fundamental and well-studied problem, in some ways a natural outgrowth of Klein’s Erlanger programme, to perceive the geometric structure on a space in terms of algebraic properties of its automorphism group. The book by Banyaga is the best reference I know for this material, in the context of “flexible” geometric structures, with big transformation groups (it is furthermore the only math book I know with a pink cover).

Left orderability is inherited under extensions. I.e. if is a short exact sequence, and both and are left orderable, then so is . Furthermore, it is a simple but useful theorem of Burns and Hale that a group is left orderable if and only if for every finitely generated subgroup there is a left orderable group and a surjective homomorphism . The necessity of this condition is obvious: a subgroup of a left orderable group is left orderable (by restricting the order), so one can take to be and the surjection to be the identity. One can exploit this strategy to show that certain transformation groups are left orderable, as follows:

**Example:** Suppose is a group of homeomorphisms of some space , with a nonempty fixed point set. If is a finitely generated subgroup of , then there is a point in the frontier of so that has a nontrivial image in the group of germs of homeomorphisms of at . If this group of germs is left-orderable for all , then so is by Burns-Hale.

**Example:** (Rolfsen-Wiest) Let be the group of PL homeomorphisms of the unit disk (thought of as a PL square in the plane) fixed on the boundary. If is a finitely generated subgroup, there is a point in the frontier of . Note that has a nontrivial image in the group of piecewise linear homeomorphisms of the projective space of lines through . Since the fixed point set of a finitely generated subgroup is equal to the intersection of the fixed point sets of a finite generating set, it is itself a polyhedron. Hence fixes some line through , and therefore has a nontrivial image in the group of homeomorphisms of an interval. By Burns-Hale, is left orderable.

**Example:** Let be the group of diffeomorphisms of the unit disk, fixed on the boundary. If is a finitely generated subgroup, then at a non-isolated point in the group fixes some tangent vector to (a limit of short straight lines from to nearby fixed points). Consequently the image of in is reducible, and is conjugate into an affine subgroup, which is left orderable. If the image is nontrivial, we are done by Burns-Hale. If it is trivial, then the linear part of at is trivial, and therefore by the Thurston stability theorem, there is a nontrivial homomorphism from to the (orderable) group of translations of the plane. By Burns-Hale, we conclude that is left orderable.

The second example does not require infinite differentiability, just , the necessary hypothesis to apply the Thurston stability theorem. This is such a beautiful and powerful theorem that it is worth making an aside to discuss it. Thurston’s theorem says that if is a finitely generated group of germs of diffeomorphisms of a manifold fixing a common point, then a suitable limit of rescaled actions of the group near the fixed point converge to a nontrivial action by translations. One way to think of this is in terms of power series: if is a group of real analytic diffeomorphisms of the line, fixing the point , then every can be expanded as a power series: . The function is a multiplicative homomorphism; however, if the logarithm of is identically zero, then if is the first index for which some is nonzero, then is an *additive* homomorphism. The choice of coefficient is a “gauge”, adapted to , that sees the most significant nontrivial part; this leading term is a character (i.e. a homomorphism to an abelian group), since the nonabelian corrections have strictly higher degree. Thurston’s insight was to realize that for a *finitely generated* group of germs of diffeomorphisms with trivial linear part, one can find some gauge that sees the most significant nontrivial part of the action of the group, and at this gauge, the action looks abelian. There is a subtlety, that one must restrict attention to *finitely generated* groups of homeomorphisms: on each scale of a sequence of finer and finer scales, one of a finite generating set differs the most from the identity; one must pass to a subsequence of scales for which this one element is constant (this is where the finite generation is used). The necessity of this condition is demonstrated by a theorem of Sergeraert: the group of germs of () diffeomorphisms of the unit interval, infinitely tangent to the identity at both endpoints (i.e. with trivial power series at each endpoint) is perfect, and therefore admits no nontrivial homomorphism to an abelian group.

Let us now return to the original question. The examples above suggest that it might be possible to find a left ordering on the group of homeomorphisms of the disk, fixed on the boundary. However, I think this is misleading. The construction of a left ordering in either category (PL or smooth) was ad hoc, and depended on locality in two different ways: the locality of the property of left orderability (i.e. Burns-Hale) and the tameness of groups of PL or smooth homeomorphisms blown up near a common fixed point. Rescaling an arbitrary homeomorphism about a fixed point does not make things any less complicated. Burns-Hale and Filipkiewicz together suggest that one should look for a structural dissimilarity between the group of homeomorphisms of the disk and of the interval that persists in finitely generated subgroups. The simplest way to distinguish between the two spaces algebraically is in terms of their lattices of closed (or equivalently, open) subsets. To a topological space , one can associate the lattice of (nonempty, for the sake of argument) closed subsets of , ordered by inclusion. One can reconstruct the space from this lattice, since points in correspond to minimal elements. However, any surjective map defines an embedding , so there are many structure-preserving morphisms between such lattices. The lattice is an -space in an obvious way, and one can study algebraic maps together with homomorphisms for which the algebraic maps respect the induced -structures. A weaker “localization” of this condition asks merely that for points (i.e. minimal elements) in the same -orbit, their images in are in the same -orbit. This motivates the following:

**Proposition:** There is a surjective map from the unit interval to the unit disk so that the preimages of any two points are homeomorphic.

*Sketch of Proof:* This proposition follows from two simpler propositions. The first is that there is a surjective map from the unit interval to itself so that every point preimage is a Cantor set. The second is that there is a surjective map from the unit interval to the unit disk so that the preimage of any point is finite. A composition of these two maps gives the desired map, since a finite union of Cantor sets is itself a Cantor set.

There are many surjective maps from the unit interval to the unit disk so that the preimage of any point is finite. For example, if is a hyperbolic three-manifold fibering over the circle with fiber , then the universal cover of a fiber is properly embedded in hyperbolic -space, and its ideal boundary (a circle) maps surjectively and finitely-to-one to the sphere at infinity of hyperbolic -space. Restricting to a suitable subinterval gives the desired map.

To obtain the first proposition, one builds a surjective map from the interval to itself inductively; there are many possible ways to do this, and details are left to the reader. qed.

It is not clear how much insight such a construction gives.

Another approach to the original question involves trying to construct an explicit (finitely generated) subgroup of the group of homeomorphisms of the disk that is not left orderable. There is a “cheap” method to produce finitely presented groups with no left-orderable quotients. Let be a group defined by a presentation, where is a word in the letters and , and is a word in the letters and . In any left-orderable quotient in which both and are nontrivial, after reversing the orientation if necessary, we can assume that . If further then , contrary to the fact that . If , then , contrary to the fact that . In either case we get a contradiction. One can try to build by hand nontrivial homeomorphisms of the unit disk, fixed on the boundary, that satisfy . Some evidence that this will be hard to do comes from the fact that the group of smooth and PL homeomorphisms of the disk are in fact left-orderable: any such can be arbitrarily well-approximated by smooth ; nevertheless at least one of the words evaluated on any smooth will be nontrivial. Other examples of finitely presented groups that are not left orderable include higher **Q**-rank lattices (e.g. subgroups of finite index in when ), by a result of Dave Witte-Morris. Suppose such a group admits a faithful action by homeomorphisms on some closed surface of genus at least . Since such groups do not admit homogeneous quasimorphisms, their image in the mapping class group of the surface is finite, so after passing to a subgroup of finite index, one obtains a (lifted) action on the universal cover. If the genus of the surface is at least , this action can be compactified to an action by homeomorphisms on the unit disk (thought of as the universal cover of a hyperbolic surface) fixed pointwise on the boundary. Fortunately or unfortunately, it is already known by Franks-Handel (see also Polterovich) that such groups admit no area-preserving actions on closed oriented surfaces (other than those factoring through a finite group), and it is consistent with the so-called “Zimmer program” that they should admit no actions even without the area-preserving hypothesis when the genus is positive (of course, admits a projective action on ). Actually, higher rank lattices are very fragile, because of Margulis’ normal subgroup theorem. Every normal subgroup of such a lattice is either finite or finite index, so to prove the results of Franks-Handel and Polterovich, it suffices to find a single element in the group of infinite order that acts trivially. Unipotent elements are exponentially distorted in the word metric (i.e. the cyclic subgroups they generate are not quasi-isometrically embedded), so one “just” needs to show that groups of area-preserving diffeomorphisms of closed surfaces (of genus at least ) do not contain such distorted elements. Some naturally occurring non-left orderable groups include some (rare) hyperbolic -manifold groups, amenable but not locally indicable groups, and a few others. It is hard to construct actions of such groups on a disk, although certain flows on -manifolds give rise to actions of the fundamental group on a plane.

Mapping class groups (also called modular groups) are of central importance in many fields of geometry. If is an oriented surface (i.e. a -manifold), the group of orientation-preserving self-homeomorphisms of is a *topological group* with the compact-open topology. The *mapping class group* of , denoted (or by some people) is the group of path-components of , i.e. , or equivalently where is the subgroup of homeomorphisms isotopic to the identity.

When is a surface of finite type (i.e. a closed surface minus finitely many points), the group is finitely presented, and one knows a great deal about the algebra and geometry of this group. Less well-studied are groups of the form when is of *infinite* type. However, such groups do arise naturally in dynamics.

**Example:** Let be a group of (orientation-preserving) homeomorphisms of the plane, and suppose that has a bounded orbit (i.e. there is some point for which the orbit is contained in a compact subset of the plane). The closure of such an orbit is compact and -invariant. Let be the union of the closure of with the set of bounded open complementary regions. Then is compact, -invariant, and has connected complement. Define an equivalence relation on the plane whose equivalence classes are the points in the complement of , and the connected components of . The quotient of the plane by this equivalence relation is again homeomorphic to the plane (by a theorem of R. L. Moore), and the image of is a totally disconnected set . The original group admits a natural homomorphism to the mapping class group of . After passing to a -invariant closed subset of if necessary, we may assume that is minimal (i.e. every orbit is dense). Since is compact, it is either a finite discrete set, or it is a Cantor set.

The mapping class group of contains a subgroup of finite index fixing the end of ; this subgroup is the quotient of a *braid group* by its center. There are many tools that show that certain groups cannot have a big image in such a mapping class group.

Much less studied is the case that is a Cantor set. In the remainder of this post, we will abbreviate by . Notice that any homeomorphism of extends in a unique way to a homeomorphism of , fixing the point at infinity, and permuting the points of the Cantor set (this can be seen by thinking of the “missing points” intrinsically as the space of *ends* of the surface). Let denote the mapping class group of . Then there is a natural surjection whose kernel is (this is just the familiar Birman exact sequence).

The following is proved in the first section of my paper “Circular groups, planar groups and the Euler class”. This is the first step to showing that any group of orientation-preserving diffeomorphisms of the plane with a bounded orbit is circularly orderable:

**Proposition:** There is an injective homomorphism .

*Sketch of Proof:* Choose a complete hyperbolic structure on . The Birman exact sequence exhibits as a group of (equivalence classes) of homeomorphisms of the universal cover of this hyperbolic surface which commute with the deck group. Each such homeomorphism extends in a unique way to a homeomorphism of the circle at infinity. This extension does not depend on the choice of a representative in an equivalence class, and one can check that the extension of a nontrivial mapping class is nontrivial at infinity. qed.

This property of the mapping class group does not distinguish it from mapping class groups of surfaces of finite type (with punctures); in fact, the argument is barely sensitive to the topology of the surface at all. By contrast, the next theorem demonstrates a significant difference between mapping class groups of surfaces of finite type, and . Recall that for a surface of finite type, the group acts simplicially on the *complex of curves* , a simplicial complex whose simplices are the sets of isotopy classes of essential simple closed curves in that can be realized mutually disjointly. A fundamental theorem of Masur-Minsky says that (with its natural simplicial path metric) is -hyperbolic (though it is not locally finite). Bestvina-Fujiwara show that any reasonably big subgroup of contains lots of elements that act on weakly properly, and therefore such groups admit many nontrivial quasimorphisms. This has many important consequences, and shows that for many interesting classes of groups, every homomorphism to a mapping class group (of finite type) factors through a finite group. In view of the potential applications to dynamics as above, one would like to be able to construct quasimorphisms on mapping class groups of infinite type.

Unfortunately, this does not seem so easy.

**Proposition:** The group is uniformly perfect.

*Proof:* Remember that denotes the mapping class group of . We denote the Cantor set in the sequel by .

A closed disk is a *dividing disk* if its boundary is disjoint from , and separates into two components (both necessarily Cantor sets). An element is said to be *local* if it has a representative whose support is contained in a dividing disk. Note that the closure of the complement of a dividing disk is also a dividing disk. Given any dividing disk , there is a homeomorphism of the sphere permuting , that takes off itself, and so that the family of disks are pairwise disjoint, and converge to a limiting point . Define to be the infinite product . Notice that is a well-defined homeomorphism of the plane permuting . Moreover, there is an identity , thereby exhibiting as a commutator. The theorem will therefore be proved if we can exhibit any element of as a bounded product of local elements.

Now, let be an arbitrary homeomorphism of the sphere permuting . Pick an arbitrary . If then let be a local homeomorphism taking to a disjoint point , and define . So without loss of generality, we can find where is local (possibly trivial), and . Let be a sufficiently small dividing disk containing so that is disjoint from , and their union does not contain every point of . Join to by a path in the complement of , and let be a regular neighborhood, which by construction is a dividing disk. Let be a local homeomorphism, supported in , that interchanges and , and so that is the identity on . Then is itself local, because the complement of the interior of a dividing disk is also a dividing disk, and we have expressed as a product of at most three local homeomorphisms. This shows that the commutator length of is at most , and since was arbitrary, we are done. qed.

The same argument just barely fails to work with in place of . One can also define dividing disks and local homeomorphisms in , with the following important difference. One can show by the same argument that local homeomorphisms in are commutators, and that for an arbitrary element there are local elements so that is the identity on a dividing disk; i.e. this composition is *anti-local*. However, the complement of the interior of a dividing disk in the plane is not a dividing disk; the difference can be measured by keeping track of the point at infinity. This is a restatement of the Birman exact sequence; at the level of quasimorphisms, one has the following exact sequence: .

The so-called “point-pushing” subgroup can be understood geometrically by tracking the image of a proper ray from to infinity. We are therefore motivated to consider the following object:

**Definition:** The *ray graph* is the graph whose vertex set is the set of isotopy classes of proper rays , with interior in the complement of , from a point in to infinity, and whose edges are the pairs of such rays that can be realized disjointly.

One can verify that the graph is connected, and that the group acts simplicially on by automorphisms, and transitively on vertices.

**Lemma:** Let and suppose there is a vertex such that share an edge. Then is a product of at most two local homeomorphisms.

*Sketch of proof:* After adjusting by an isotopy, assume that and are actually disjoint. Let be sufficiently small disjoint disks about the endpoint of and , and an arc from to disjoint from and , so that the union does not separate the part of outside . Then this union can be engulfed in a punctured disk containing infinity, whose complement contains some of . There is a local supported in a neighborhood of such that is supported (after isotopy) in the complement of (i.e. it is also local). qed.

It follows that if has a bounded orbit in , then the commutator lengths of the powers of are bounded, and therefore vanishes. If this is true for every , then Bavard duality implies that admits no nontrivial homogeneous quasimorphisms. This motivates the following questions:

**Question:** Is the diameter of infinite? (Exercise: show )

**Question:** Does any element of act on with positive translation length?

**Question:** Can one use this action to construct nontrivial quasimorphisms on ?

## Amenability of Thompson’s group F?

July 6, 2009 in Commentary, Groups | Tags: Akhmedov, amenability, Dynamics, Shavgulidze, Thompson's group | by Danny Calegari | 12 comments

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 virtually orderable?

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

Spaces and Questionsremarks thatA famous question of the kind Gromov warns against is the following:

Question:Is Thompson’s group 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 for integers ) and derivatives all powers of . This group is a rich source of examples/counterexamples in geometric group theory: it is finitely presented (in fact ) 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 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 -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 is amenable. Interestingly enough, Shavgulidze’s argument does

notapply 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 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 . In more detail, let denote the group of diffeomorphisms of the interval of regularity at least for each , and let denote the Banach space of continuous functions on the interval that vanish at the origin. Define by the formula . The space can be equipped with a natural measure — theWiener measureof variance , and this measure can be pulled back to by , which is thought of as a topological space with the topology. Shavgulidze shows that the left action of on quasi-preserves this measure. Here the Wiener measure on is the probability measure associated to Brownian motion (with given variance). A “sample” trajectory from is characterized by three properties: that it starts at the origin (i.e. ), that is it continuous almost surely (this is already implicit in the fact that the measure is supported on the space and not some more general space), and that increments are independent, with the distribution of equal to a Gaussian with mean zero and variance . 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 diffeomorphisms of the interval. A bounded function on determines a continuous bounded function on (for ) by a certain convolution trick, using both the group structure of , and its discreteness in . Roughly, given an element , the set of elements of whose (group) composition with is uniformly bounded in the norm is finite; so the value of is obtained by taking a suitable average of the value of on this finite subset of . This reduces the problem of the amenability of to the existence of a suitable functional on the space of bounded continuous functions on , 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 on — 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 — is also crucial for many other natural problems in -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 that is discrete in the 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 contains a discrete nonabelian free subgroup.Sketch of Proof:First, decompose the interval 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 at the endpoints. This can certainly be accomplished; for concreteness, choose the action so that for each subinterval there is a point in the interior of 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 at some point. In more detail: enumerate the pairs of words somehow where each is a pair of words in the generators, and modify the action on the subinterval so the words in differ by at least in the norm on the interval . 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 . Since there are only finitely many pairs of words, both of which are of bounded length (for any given bound), when is sufficiently big, one of the words , has length at least where goes to infinity as goes to infinity. Without loss of generality, we can order the pairs so that is the “long” word.

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

It is interesting to observe that this construction, while , is not for any . For big , we have whereas . Introducing a “bump” which modifies the derivative by in a subinterval of size 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:In light of this, it would seem to be reasonable to consider the question of whether is amenable as wide open.

(Update 9/21/2012):Justin Moore has posted a preprint on the arXiv claiming to prove amenability of . 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.