SUMMARY of RESEARCH Laurent Bartholdi's Recent Research Has

SUMMARY OF RESEARCH

LAURENT BARTHOLDI

Laurent Bartholdi’s recent research has focused on constructions of groups and algebras with unusual, striking properties, and on the links between such constructions and other areas of mathematics such as complex dynamical systems. He has attempted to build up a conceptual platform within which to study self-similar algebraic objects, rather than to study individual examples. This uni- fying theme is expressed in the language of finite-state automata; specific outcomes of the theory are described in §2 (Thurston equivalence and a solution to Hub- bard’s question), §3 (Growth, and a counterexample a conjecture by Grigorchuk), §4 (Amenability of groups and a solution to Ceccherini-de la Harpe-Grigorchuk’s question), §5 (Lie algebras, and a counterexample to a conjecture by Zelmanov), and §6 (Thinned algebras, and a counterexample to a conjecture by Goodearl). He has also strived to apply group theory to the solution of problems in other areas of mathematics; some of this research is summarized in §7. He has also contributed to the theory of random walks on graphs, see §8. More recently, he has been interested in automorphisms of free groups, answering some questions and disproving some conjectures by Andreadakis, see §9. He has also developed the theory of amenability for associative algebras, leading to the construction of non-amenable residually finite groups with striking properties (such as torsion), see §10.

1. A brief introduction to automata An automaton, sometimes called a Mealy machine, is a device with finite memory. Let us imagine it to be started in a given memory state, at the beginning on a tape. It then reads an input symbol from the tape and, depending on that symbol and on its memory state, chooses an output symbol to print and a new memory state to enter; it then repeats the process on the next input symbol from the tape. An automaton, with chosen initial state, therefore defines a transformation on the set of tapes (= set of sequences over a given alphabet). Given a set of automata, each with chosen initial state, and such that all the transformations are invertible, we consider the group generated by these automata. Many fundamental questions in group theory were answered using such constructions: a simple solution to Burnside’s problem by Aleshin [21], to Milnor’s problem by Grigorchuk [28], etc. 1.1. FR Package. In studying these examples of groups and algebras, a great deal of experimentation is necessary. Although the experiments usually do not appear explicitly in final results and publications, they are used to obtain evidence on the structure of groups, algebras and their identities. In particular, the solution to the next §B.1.2 was obtained experimentally before being proved.

Date: May 13, 2007. 1 2 LAURENT BARTHOLDI

Laurent Bartholdi has developed a sophisticated computer package, based on the computer algebra system Gap, to aid such experimentation. This computer environment can perform calculations on infinite groups generated by automata, starting solely from the automata’s definitions. The theoretical results mentioned below have both been a consequence of and a stimulus for the development of the FR package. The package is freely available through the Internet. He expects it to have a significant impact on the field.

2. Thurston equivalence of topological polynomials This involves an exciting interplay between group theory and complex dynamics, that has already solved an old problem, by bringing new and unexpected structures to light. 2 Consider the “rabbit” polynomial fR(z) ≈ z +(−0.1226+0.7449i), whose critical point 0 is on a periodic orbit of length 3. Up to affine transformations, there are exactly two other polynomials with same action on post-critical points (with the 2 same ramification graph), called the “corabbit” fC ≈ z + (−0.1226 − 0.7449i) and 2 the “airplane” fR ≈ z − 1.7549. Furthermore, by a result of Thurston, every branched covering with same ramification graph is equivalent to precisely one of fR, fC , fA. Consider now a Dehn twist T of C about the two non-critical values of the m fR-orbit of 0. The map T fR is again a branched covering, and it has the same ramification graph as fR; therefore it is equivalent (i.e., conjugate up to homotopies) to one of fR, fC , fA. Which one? This question was asked by Hubbard in the early 1990s. In [14], Laurent Bartholdi, in collaboration with Volodya Nekrashevych, gives the following answer: P∞ i Write m in base 4, as m = i=0 mi4 with mi ∈ {0, 1, 2, 3} and almost all mi = 0 if m is non-negative, and almost all mi = 3 if m m is negative. If one of the mi is 1 or 2, then T fR is equivalent to fA. Otherwise, it is equivalent to fR if m is non-negative, and to fC if m is negative. The method of proof uses the “iterated monodromy groups” developed by Nekra- shevych to study both fR and an associated map on the moduli space of polynomials with prescribed post-critical behaviour. These “iterated monodromy groups” are groups generated by automata, with as many memory states as there are post-critical points for the map. An extended study of the iterated monodromy groups associated with quadratic polynomials appears in [17], by Bartholdi and Nekrashevych.

3. Growth of groups Let G be a ﬁnitely generated group, generated by a set S. The growth of G with respect to S is the function −1 γ(n) = #{g ∈ G : g = s1 ··· sn for some si ∈ {1} ∪ S ∪ S }. This function naturally depends on the choice of S, but not in an essential manner. More precisely, if we declare for every K > 1 than the functions γ(n), γ(Kn) and all functions pointwise comprised between them are equivalent, then the equivalence class of γ is independent of S. SUMMARY OF RESEARCH 3

This growth function is an extremely interesting invariant. Thus, for example, a famous result by Gromov asserts that γ is equivalent to a polynomial function precisely if G contains a ﬁnite-index nilpotent subgroup. Non-abelian free groups, and more generally Gromov-hyperbolic groups, all have exponential growth. Milnor asked whether there exist groups whose growth function is larger than any polynomial but smaller than any exponential function. For instance, this cannot happen in the realm of linear groups. Grigorchuk constructed an example of such a group, and more precisely obtained the estimates √ n n0.991 e - γ(n) - e . He conjectured that the lower bound is sharp. Laurent Bartholdi improved the upper and lower bounds to

n0.515 nα e - γ(n) - e , with α = log(2)/ log(2/η) =∼ 0.767, where η is the positive root of X3 + X2 + X − 2. He thus disproved Grigorchuk’s conjecture [1,5]. In another direction, let G be a group whose growth function γ(n) is exponential. Deﬁne then the entropy of G with respect to S, log γ(n) λ (G) = lim . S n

Thus G has exponential growth if and only if λS(G) > 0 for some, or for any, generating set of G. Gromov asked in [29] “Do there exist groups for which λS(G) > 0 for all generating sets S, but nevertheless infS λS(G) = 0?”. Such groups are called groups of non-uniform exponential growth. Again, they cannot exist in the realm of matrix groups. Wilson announced in 2002 the construction of such a group. Laurent Bartholdi presents in [8] a similar, but much simpler construction.

4. Amenability of groups This fundamental notion, which impinges on much of modern mathematics, was introduced by von Neumann [38] in connection to his study of the Banach-Tarski paradox (which he reduced to a group-theoretic statement). A discrete group G is amenable if it admits a mean, i.e. a function m : P(G) → [0, 1] that is finitely additive (m(A t B) = m(A) + m(B)), translation-invariant (m(Ag) = m(A)) and normalized (m(G) = 1). Some groups are known to be amenable: abelian groups; extensions, quotients, subgroups and direct limits (ascending unions) of amenable groups. These groups are called elementarily amenable groups. pn Groups whose growth function (see §3) is subexponential ( γS(n) → 1) for all finite S ⊆ G, then G is amenable. Groups obtained from groups with subexponential growth using extensions, quotients, subgroups and direct limits are called subexponentially amenable groups. Other groups, such as free groups, or more generally groups with free subgroups, are not amenable (this is the origin of the Banach-Tarski paradox). M. Day asked [25] whether all amenable groups are elementarily amenable. This was infirmed by Grigorchuk, and led to the following question, explicitly stated by Ceccherini, Grigorchuk and de la Harpe [24]: 4 LAURENT BARTHOLDI

“Subexponentially amenable groups strictly contain elementarily amenable groups. Are there amenable groups that are not subexponentially amenable?” In [10], Laurent Bartholdi, in collaboration with Balint Virag, answers this question positively by exhibiting such a group. It is a group generated by very small automata (3 memory states).

5. Lie algebras A standard construction, due to Magnus, associates a Lie algebra with an arbi- trary (not necessarily Lie) group. Given a group G, define its lowar central series by γ1(G) = G and γn+1(G) = [γn(G),G]. Then γn(G)/γn+1(G) is an abelian group, and we set M L (G) = γn(G)/γn+1(G); n≥1 it is a graded Z-module. Furthermore, a simple calculation shows that [γm(G), γn(G)] ⊆ γm+n(G) ; thus, L (G) is a Lie algebra for the product defined on homogeneous components by [Xγm+1(G), Y γn+1(G) = [X,Y ]γm+n+1(G). The Lie algebra L (G) is finite-dimensional only when G is a finitely generated nilpotent group. The next interesting case occurs then if the rank of γn(G)/γn+1(G) is bounded ; then G is said to have finite width. For instance, analytic groups have this property. Zelmanov studied this property in a large class of groups [32], and was thus led to conjecture that any pro-p group of finite width is analytic [42]. In [4], Laurent Bartholdi and Rostislav Grigorchuk, compute the Lie algebra structure of L (G) for the Grigorchuk group G of intermediate growth (see §3), and they show that γn(G)/γn+1(G) has rank at most s 3 ; this disproves Zelmanov’s conjecture. Laurent Bartholdi then determined in [11] the structure of the Lie algebra of close parent of Grigorchuk’s group (the “Gupta-Sidki group”), and shows that it does not have finite width (the ranks grow polynomially in n). He also showed that this is the general case. He then used the structure of the Lie algebra of Grigorchuk’s group G to deter- mine the lattice of normal subgroups of G. He shows in particular that this lattice is self-similar, and that there are asymptotically nlog2(3) normal subgroups of index 2n.

6. Thinned algebras The concept of automata may be generalized to a linear setting; in this way, a linear automaton is used to generate an associative algebra. This important notion has already led to the construction of striking, previously-unknown associative algebras. Laurent Bartholdi develops in [12] the theory of “branch algebras”. They are infinite-dimensional associative algebras A that are isomorphic, up to taking sub- rings of finite codimension, to a matrix ring over themselves. More precisely, they admit an embedding φ : A → Md(A) and a finite-codimensional ideal K such that φ(K) contains Md(K). In particular, for every field k he constructs a k-algebra A which • is finitely generated and infinite-dimensional, but has only finite-dimensional quotients; SUMMARY OF RESEARCH 5

• has a subalgebra of finite codimension, isomorphic to M2(A); • is prime; • has quadratic growth, and therefore Gelfand-Kirillov dimension 2; • is recursively presented; • satisfies no identity; • contains a transcendental, invertible element; • is semiprimitive if k has characteristic 6= 2; • is graded if k has characteristic 2; • is primitive if k is a non-algebraic extension of F2; • is graded nil and Jacobson radical if k is an algebraic extension of F2. It was conjectured by Goodearl that if A is an algebra over a field k and has Gelfand-Kirillov dimension 2, then its Jacobson radical is nil. The algebra A above therefore provides a counterexample to that conjecture.

7. Applications of Group Theory Laurent Bartholdi has a strong strong belief in the ability of group theory to solve problems throughout mathematics. For example, he has been interested in their use to study the quantum Yang-Baxter (qYB) equation. Brieﬂy said, the qYB equation in an associative algebra A is the equation r12r13r23 = r23r13r12 expressed for an element r ∈ A ⊗ A. It may be ‘quantized’ into a group relation; the group QYB deﬁned by that relation is then a universal object in the following sense: monoid homomorphisms from QYB to the tensor algebra T (A) are in bijection with solutions of the qYB equation. In [13], Laurent Bartholdi and coauthors have used the notion of ‘non-positive curvature’ to study the group QYB, and explicitly describe its cohomology.

7.1. Group theory and the psychology of intelligence. Group theory is fundamental in Piaget’s work. In [19], Laurent Bartholdi, in collaboration with Fabrice Liardet, interpreted Piaget’s discoveries in the light of group theory, and discovered strong bonds between these domains. The impact of group theory of piagetian thought may be summed up as follows: the main step in the development of a child’s intelligence is the mental construction of “operations”; Piaget calls this the “operatory step”. Structuring the world in terms of reversible operations, which can be composed (with an associative product) naturally leads to the notion of conservation (for instance conservation of an object when it disappears from sight, or of an amount of liquid as it is poured from one vessel to another). It is then natural that group-theoretical properties are a sure indicator of the development of a child.

7.2. Lamps and ﬁnite ﬁelds. The following problem appeared in the 1992 Inter- national Mathematical Olympiad:

¡¡Let n > 1 be an integer. There are n lamps L1,...,Ln−1 placed around a circle. Each lamp is either OFF, or ON. One performs a sequence S0,S1,... of operations. The operation Sj changes only lamp Lj, as follows: if Lj−1 is ON, then Sj changes the state of lamp Lj (from ON to OFF and from OFF to ON), while if Lj−1 is OFF, then Sj does not change lamp Lj. Lamps are numbered modulo n, and are initially all ON. Show: 6 LAURENT BARTHOLDI

(1) that there exists an integer M(n) such that after M(n) steps all lamps are again ON; (2) that if n is of the form 2k then all lamps are again ON after n2 − 1 steps; (3) that if n is of the form 2k +1 then all lamps are again On after n2 − n + 1 steps. In the paper [3], Laurent Bartholdi solves this problem by relating it to the factor- n ization of Φn = X + X + 1 in caracteristic 2. Supposing M(n) minimal, he shows that M(2n − 1) divides 2M(n) − 1, and conjectures, based on numerical evidence, that this is in fact an equality.

7.3. Maximal unramified 3-extensions. In the paper [15], Laurent Bartholdi and Michael Bush construct a family of finite 3-groups, which are good candidates to be the Galois groups of maximal unramified 3-extensions above a number field. Their proof is based on the identification of these groups (whose presentations arise from number-theoretic considerations) with certain quotients of subgroups of SL2(Z3). The interest in this research comes from the famous open question “Is there a bound to the height of a finite p-tower class?”. The answer would be negative if they could prove that all groups in their family are Galois groups. They prove it for the first two cases.

8. Random walks on graphs Let G be a graph, given by a collection of vertices and edges. A random walk on G is a random process described as follows: a particle is located on a vertex of G; it chooses randomly an edge out of that vertex, and moves to the other side of the edge; it then starts again from the new edge. The random walk is simple if the outgoing edges are chosen uniformly and independently at random. It is without backtracking if the outgoing edge may not be the edge on which the particle just arrived. One is in particular interested in return probabilities. These are the probabilities n px,y that, starting at vertex x, a particle reaches vertex y after n steps. In a n connected graph, the asymptotics of px,x are independent of x, and are of the form ρn where ρ is the `2 norm of the graph’s adjacency operator. P n n The classical Green function is the generating series Gx,y(t) = px,yt . If the graph G is regular, meaning that each vertex has a constant number d of neighbours, n n then px,yd is the number of paths from x to y of length n. The function Gx,y(dt) is thus a power series with integral coefficients, counting the number of paths from x to y. For a path in a graph G, define its backtracking count as the number of positions along the path at which it continues on the edge it just arrived with. One can then define a power series Fx,y(u, t) as the sum, over all paths γ from x to y, of length(γ) backtracking count(γ) t u . We have Fx,y(1, t) = Gx,y(t). SUMMARY OF RESEARCH 7

Laurent Bartholdi proves in [2] the following functional equation, for a d-regular graph G:

t F (1 − u, t) Gx,y 1+u(d−u)t2 (†) x,y = . 1 − u2t2 1 + u(d − u)t2 This functional equation is very useful to count paths in certain graphs; in particular, its full power, substituting for u a speciﬁc power series in t, is used in [2]. It is used by Laurent Bartholdi and Tullio Ceccherini-Silberstein in [6, 7] to obtain estimates on the asymptotics of the simple random walk on the Cayley graph of surface groups. An improved estimate appears in [9], using (†) multiple times to count structures (called “cacti”), giving a lower bound on the spectral radius.

8.1. ζ functions. In relation to algebraic geometry and number theory questions [30], Ihara considered a power series ζ counting “cycles” in a graph — these are closed paths with no backtracking, considered up to cyclic permutation of their edges, which are not proper powers of another path. Bass [23] showed how to express this power serios in terms of the graph’s adjacency matrix. Still in the paper [2], Laurent Bartholdi constructs a two-variable ζ function (one variable counts length, the other the number of backtrackings), and shows that this function can be obtained quits simply from the graph’s adjacency matrix, or from G∗,∗(u, t). This generalized ζ function has then been extensively studied [31,33–37, 39, 40].

9. The automorphism group of a free group Let F denote a free group of rank r. The group-theoretical structure of the automorphisms group A of F is probably exceedingly diﬃcult to describe, but A may be ‘graded’, following Andreadakis [22], into a more manageable object. Let again (γn(F )) denote the lower central series of F . Since γn(F ) is invariant under all automorphisms of F , there is then a natural map aut(F ) → aut(F/γn+1(F )), whose kernel we denote by An. Then A0/A1 = GLr(Z), and An/An+1 are ﬁnite- rank free Z-modules; furthermore, [An,Am] ⊆ Am+n, and therefore M L = An/An+1 n≥1 has the structure of a Lie algebra. Let, by comparison, Fb denote the free pronilpotent group of rank r; it is the limit of the F/γn(F ). Let again Abn denote the kernel of the natural map aut(Fb) → L aut(F/γn+1(F )). Then Ab0/Ab1 = GLr(Zb) and M = n≥1 Abn/Abn+1 is also a Lie algebra. There is a natural map L → M , which however does not have dense image. The following problems appear naturally: (1) Describe the image of L in M . (2) Relate An to the lower central series (γn(A1)) of A1. (3) Compute the rank of Ln = An/An+1.

Ad (1), Andreadakis observes that L1 = M1 and L2 = M2, while L3 6= M3. Ad (2), Andreadakis conjectures [22, page 253] that An = γn(A1), and proves his assertion for r = 3, n ≤ 3 and for r = 2. 8 LAURENT BARTHOLDI

Ad (3), Andreadakis proves

r X (n+1)/d rank Mn = µ(d)r , n + 1 d|n+1 where µ denote the M¨obius function, and computes for r = 3 the ranks rank(Ln) = 9, 18, 44 for n = 2, 3, 4 respectively. Laurent Bartholdi proves in [20] the following:

• For all r, n we have γn(A1) ≤ An, and An/γn(A1) is a ﬁnite group. More- over, p An = γn(A1), k that is, An is the set of all g ∈ A such that g ∈ γn(A1) for some k 6= 0. On the other hand, for r = 3 and n = 7 we have An/γn(A1) = Z/3Z. Therefore, Andreadakis’s conjecture is false, but barely so. • If r ≥ n + 2, then we have the rank formula

r X (n+1)/d 1 X n/d rank Ln = µ(d)r − φ(d)r , n + 1 n d|n+1 d|n where φ denotes the Euler totient function. In fact, the arguments show that there is a mistake in Andreadakis’s calculations for r = 3, which should be corrected to rank(L4) = 43. These results are proven by a detailed understanding of the Ln and the Mn as GLr(Z)-modules.

10. Amenable algebras One may define amenability for an associative algebra as the existence of almost- invariant vector subspaces, in analogy to Følner’s definition for groups. Laurent Bartholdi shows in [16] that a group is amenable if and only if its group algebra is amenable; this was only known in the easy direction “Amenable group implies amenable group algebra” [26]. Let G be a group, and let (γn(G)) denote its lower central series (see §5). Laurent Bartholdi proves in [18] the following result: if G is finitely generated and amenable, then the rank of γn(G)/γn+1 grows subexponentially in n. As a corollary, this answers positively an open question by de la Harpe, who asked “Do there exist residually-finite, finitely generated, non-amenable torsion groups?”. This also proves a conjecture by Vershik [41]. More geneally, Laurent Bartholdi shows that Golod-Shafarevich groups [27] are not amenable. Since Golod constructed torsion residually-p Golod-Shafarevich groups, de la Harpe’s question has a positive answer.

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