Posted by: matheuscmss | March 26, 2013

Second Bourbaki seminar of 2013

Last Saturday (March 23, 2013), the second Bourbaki seminar of this year took place at amphithéâtre Hermite of Institut Henri Poincaré (as usual), and the following topics were discussed:

Once more the speakers did a great job in explaining these topics to an audience of non-experts, and, for this reason, I decided to make a post about one of these talk.

Contrary to last time, it was “easy” for me to choose which topic to pick: given my tastes, I had to choose between Ore’s conjecture and ending laminations, and I opted for Ore’s conjecture because the website Images de Mathématiques made available an excellent article (in French) by F. Guéritaud with a guided tour (with plenty of beautiful pictures!) around the works of Masur-Misnky and Brock-Canary-Minsky.

So, below I will transcript my notes of G. Malle’s talk about Ore’s conjecture. As usual, the eventual mistakes in what follows are my entire responsibility.

1. Introduction

Let {G} be a group. Given {x,y\in G}, we denote by {[x,y]:=x^{-1}y^{-1}xy} the commutator of {x} and {y}, and we define the commutator subgroup {[G,G]} the subgroup of {G} generated by the elements {[x,y]}, for {x,y\in G}. In group theory, {[G,G]} is used to define the derived series helping to distinguish certain “categories” of groups such as perfect groups (when {G=[G,G]}), solvable groups, etc.

By definition, {\{[x,y]:x,y\in G\}\subset [G,G]}, but, in general, this inclusion is not an equality: for example, there are two (non-isomorphic) groups of order {96} such that the set of commutators {\{[x,y]:x,y\in G\}} is different from {[G,G]} (cf. this article of R. Guralnick).

In a more positive tone, N. Ito and O. Ore (independently) showed that:

Theorem 1 (Ito, Ore (1951)) For {G=S_n} (symmetric group) and {G=A_n} (alternating group), we have the equality {\{[x,y]:x,y\in G\}=[G,G]}.

Furthermore, O. Ore said in his paper that “it is possible that a similar theorem holds for any simple group of finite order, but it seems that at present we do not have the necessary methods to investigate the question”. For this reason, the question of determining whether the previous theorem extends to all finite simple groups became Ore’s conjecture.

In 2010, M. Liebeck, E. O’Brien, A. Shalev and P. Tiep showed in this paper here that Ore’s conjecture is true:

Theorem 2 (Liebeck-O’Brien-Shalev-Tiep (2010)) If {G} is a non-abelian finite simple group, then every {g\in G} is a commutator.

As G. Malle pointed out to us, the proof of Ore’s conjecture by LOST (Liebeck-O’Brien-Shalev-Tiep :) ) “confirms” O. Ore’s prediction that the tools to attack this problem were not available at the time he wrote his paper. Indeed, LOST’s proof of Ore’s conjecture uses:

Concerning the statement of Ore’s conjecture/LOST theorem, let us notice that the result is simply not true for general non-simple groups {G}. For instance, R. Guralnick showed that if {U} and {H} are two finite groups such that {U} is Abelian, {|U|>2}, and {|[H:H]|>2}, then the regular wreath product {G=U\wr H} doesn’t satisfy the equality {[G,G]=\{[x,y]: x,y\in G\}}. (Here, we recall that the wreath product {U\wr H} is formed by taking the direct product {\prod_{h\in H} U_h} of several copies {U_h} of {U} indexed by the elements {h} of {H} and by observing that {H} acts [regularly by left multiplication] on this direct product by permuting the indices, so that it makes sense to talk about {U\wr H} the semi-direct product of {\prod_{h\in H} U_h} and {H}). Also, it is known that the smallest example of a perfect group not fitting the conclusion of Ore’s conjecture/LOST theorem is an extension of {C_2^4} (i.e., an elementary Abelian group of order {2^4}) by the alternating group {A_5}.

A question related to Ore’s conjecture and normally attributed to J. G. Thompson is the following one:

Conjecture (J. G. Thompson). Let {G} be a finite non-Abelian simple group. Then, there exists a conjugacy class {C} such that {C^2:=\{xy: x,y\in C\}} equals to {G} ({C^2=G}).

Note that Thompson’s conjecture implies Ore’s conjecture. Indeed, if {G=C^2}, then any {g\in G} has the form {g=xy} with {x,y\in C}. In particular, whenever {u\in C}, its inverse {u^{-1}} also belongs to {C}: by hypothesis, we can write the identity element {1\in G} as a product {1=zw} for some {z,w\in C}, so that there is an element {z\in C} whose inverse {w=z^{-1}} also belongs to {C}, and, a fortiori, the same is true for all elements {u} of {C} (as they are conjugated to {z}). Therefore, if we write {g\in G} as {g=xy} with {x,y\in C}, then, since {y^{-1}\in C} (as we just saw), we know that there exists {h\in G} such that {x=h^{-1}y^{-1}h} and, thus, {g=xy=h^{-1}y^{-1}hy=[h,y]\in [G,G]}.

Actually, prior to LOST theorem, the proofs of Ore’s conjecture for certain families of groups proceeded by showing that Thompson’s conjecture is true for those families (and this is why we’re mentioning Thompson’s conjecture here). However, this is not the way that LOST’s arguments will work and, in particular, Thompson’s conjecture is still open for the general finite simple group.

2. Direct calculations

The “essential” case in LOST’s proof is that of {G} of Lie type, e.g.,

\displaystyle G=PSL_n(\mathbb{F}_q), PSp_{2n}(\mathbb{F}_q),\dots, E_8(\mathbb{F}_q).

Indeed, the idea is that for the “remaining” groups (such as the 26 sporadic ones), one can verify Ore’s conjecture by “hands” (or, more precisely, using computers).

Before discussing the case of groups of Lie type, let us consider first the case of simple algebraic groups (that is, the “continuous” analogs of finite groups of Lie type). In this direction, one has the following theorem of S. Pasiencier and H. Wang (over the complex numbers) and R. Ree (over arbitrary algebraically closed fields):

Theorem 3 (Pasiencier-Wang, Ree (1964)) Let {G} be a semisimple linear algebraic group over an algebraically closed field. Then, every {g\in G} is a commutator.

Proof: Given {g\in G}, we have that {g} belongs to some Borel subgroup {B} (by a result of Borel). Denote by {B=UT} Levi’s decomposition of {B} with {U=R_u(B)} the (unipotent) radical of {B} and {T} a maximal torus.

In this notation, the proof of the theorem is based on the following two facts:

  • (1) For all {s\in T}, there exists {t\in T} a regular element (i.e., an element whose centralizer {C_G(t)} is {T}) and an element {x\in N_G(T)} of the normalizer of {T} such that {x^{-1}tx=ts}.
  • (2) if {t\in T} is regular, then {tU} is a single {B}-conjugacy class.

The first item is due to Kostant and the second item is not hard to deduce (one shows by induction on the central series of {U} that the map {u\mapsto [t,u]} from {U} to {U} is bijective when {t} is regular).

In any case, given {g\in G} with Jordan decomposition {g=su}, {s\in T} and {u\in U}, we use item (1) to select {t\in T} regular and {x\in N_G(T)} with {x^{-1}tx=ts}. Since {t\in T} is regular, we have that {ts\in T} is also regular, and, thus, by item (2), there exists {b\in B} such that {tsu=b^{-1}tsb}. It follows that

\displaystyle g=su=t^{-1}tsu=t^{-1}(b^{-1}tsb)=t^{-1}b^{-1}(x^{-1}tx)b=[t,xb],

so that the proof of the theorem is complete. \Box

It is tempting to try to adapt this proof to the case of finite groups of Lie type, but, as it turns out, this is not so easy because it is not true that all elements lie in some Borel subgroup and the analog of item (1) above is simply false.

Nevertheless, E. Ellers and N. Gordeev obtained in these papers here the following result about Gauss decompositions of elements of finite simple groups of Lie type:

Theorem 4 (Ellers-Gordeev (1996)) Let {G} be a finite simple group of Lie type. Denote by {T} a maximally split torus inside a Borel subgroup {B} of {G} with Levi decomposition {B=UT}. Consider {B^-} the opposite Borel subgroup and let {B^-=U^-T} be its Levi decomposition. Fix {t\in T}. Then, for any {1\neq g\in G}, there exists {x\in G} such that {xgx^{-1}=u_1 t u_2} with {u_1\in U^-} and {u_2\in U}.

The next two corollaries show why this theorem is helpful in our discussion:

Corollary 5 If {C_1} and {C_2} denote the {G}-conjugacy classes of two regular elements {t_1, t_2\in T}, then {G=C_1C_2\cup\{1\}}.

Proof: By Ellers-Gordeev theorem, if {1\neq g\in G}, then, by taking {t=t_1t_2}, we have

\displaystyle xgx^{-1}=u_1t_1t_2u_2

On the other hand, by applying item (2) in the proof of Theorem 3 ({tU} is a single {B}-conjugacy class when {t} is regular), we deduce the existence of elements {v_1} and {v_2} such that

\displaystyle u_1t_1=v_1t_1v_1^{-1}\in C_1 \quad \textrm{and} \quad t_2u_2=v_2t_2v_2^{-1}\in C_2

It follows that

\displaystyle xgx^{-1}= (v_1t_1v_1^{-1})(v_2t_2v_2^{-1})\in C_1C_2,

so that the proof of the corollary is complete. \Box

This corollary provides the following criterion for the validity of Thompson’s and Ore’s conjectures for finite simple groups of Lie type:

Corollary 6 If there exists a regular element {t\in T}, then Thompson’s (and, a fortiori, Ore’s) conjecture is true.

Proof: By taking {t_1=t_2=t} in the previous corollary, we have that any {1\neq g\in G} belongs to {C^2} where {C} is the conjugacy class of {t}. \Box

Using this corollary (among other arguments), E. Ellers and N. Gordeev showed in this paper here the following result:

Theorem 7 (Ellers-Gordeev (1998)) Let {G} be a finite simple group of Lie type over a finite field {\mathbb{F}_q} with {q\geq 9}. Then, Thompson’s (and Ore’s) conjecture is true.

In view of this theorem, Liebeck-O’Brien-Shalev-Tiep showed their theorem (Ore’s conjecture) by developing a character theory method for the remaining cases that we are going to discuss in the next section.

3. Character theory

Denote by {Irr(G)} the set of irreducible (complex) characters of {G}. A classical lemma due to Frobenius gives the following criterion to recognize whether an element {g\in G} is a commutator in terms of the characters of {G}:

Lemma 8 (Frobenius) Let {G} be a finite group. Then, {g\in G} is a commutator if and only if

\displaystyle \sum\limits_{\chi\in Irr(G)}\frac{\chi(g)}{\chi(1)}\neq0

Note that this criterion is suitable for computer calculations: indeed, if we know the character table of {G}, we can ask a computer to determine the sum {\sum\limits_{\chi\in Irr(G)}\frac{\chi(g)}{\chi(1)}} in order to recognize what elements of {g\in G} are commutators. In particular, this approach works to verify Ore’s conjecture for the 26 sporadic groups (because their character tables are known).

LOST’s idea consists into applying Frobenius lemma in the following way. Firstly, we separate the contribution of the trivial character {1_G} from other characters

\displaystyle \left|\sum\limits_{\chi\in Irr(G)}\frac{\chi(g)}{\chi(1)}\right|\geq 1-\left|\sum\limits_{\chi\neq 1_G}\frac{\chi(g)}{\chi(1)}\right|.

Secondly, we use the classical bound {|\chi(g)|^2\leq |C_G(g)|} (coming from Schur orthogonality relations) to see that

\displaystyle \left|\sum\limits_{\chi\in Irr(G)}\frac{\chi(g)}{\chi(1)}\right|\geq 1-\sqrt{C_G(g)}\sum\limits_{\chi\neq 1_G}\frac{1}{\chi(1)}

In particular, the desired sum is not zero if the “error term” in the right-hand side is small, that is, if the conjugacy class {C_G(g)} is small and/or the dimensions {\chi(1)} of non-trivial irreducible characters {\chi} are large.

In order to make this approach work, LOST show the following “Cauchy-Schwarz inequality” controlling the contribution of (non-trivial) irreducible characters {\chi} with “large” dimension {\chi(1)}:

Lemma 9 Let {G} be a finite group with {k_G} conjugacy classes. Then, for all {g\in G} and {N>0},

\displaystyle \sum\limits_{\chi(1)\geq N}\frac{|\chi(g)|}{\chi(1)}\leq \frac{\sqrt{k_G |C_G(g)|}}{N}

Of course, a preliminary question before applying this lemma is: what is the size of {k_G}? In general, if the finite simple group of Lie type {G} has the form {G=G_r(\mathbb{F}_q)} where {r} denotes its rank, then it is known that {k_G} is bounded by a polynomial in {q} of degree {r}. For example, for {G=SL_r(\mathbb{F}_q)}, J. Fulman and R. Guralnick proved that

\displaystyle k_G=k(SL_r(\mathbb{F}_q))\leq \frac{q^r}{q-1}+q^{(r/2)+1}

Once we know how to control {k_G}, the next preliminary question concerns lower bounds on the dimensions of non-trivial irreducible characters (because this gives an idea of the size of the integer {N} for which we will apply the lemma). Here, the Lusztig’s classification of irreducible characters comes into play and one often founds that the following gap phenomenon: there are a few non-trivial irreducible characters close to the smallest possible dimensions and all other non-trivial irreducible characters have dimension at least the square of the smallest possible dimensions. For example, for the symplectic groups {G=Sp_{2n}(\mathbb{F}_q)}, it is possible to show that (cf. this paper of P. Tiep and A. Zalesski):

Lemma 10 Let {G=Sp_{2n}(\mathbb{F}_q)} with {q} odd and {n\geq 2}. Let {\chi\in Irr(G)-\{1_G\}} be a non-trivial irreducible character of {G}. Then:

  • either {\chi(1)=(q^n\pm1)/2} and {\chi} is one of the (four) so-called Weil characters of {G};
  • or {\chi(1)\geq (q^n-1)(q^{n-1}-1)/2(q+1)}.

Proof: The lower bound on {\chi(1)} is not hard to establish: we can embed {SL_2(\mathbb{F}_{q^n})=Sp_2(\mathbb{F}_{q^n})} into {Sp_{2n}(\mathbb{F}_q)} by considering {\mathbb{F}_{q^n}} as a {n}-dimensional vector space over {\mathbb{F}_q}; since {SL_2(\mathbb{F}_{q^n})} has smallest degrees {(q^{n}\pm1)/2}, we get the desired lower bound.

On the other hand, the “gap” result is hard to get: for symplectic groups, one can consult this paper for an elementary proof, but for other types one has to use the full Lusztig’s classification. \Box

After answering the preliminary questions on the sizes of {k_G} and {N}, we are ready to get back to Ore’s conjecture. By using the information on {k_G} and {N}, LOST use the estimate on {k_G} and the “gap” phenomenon above to safely concentrate on the few “Weil characters” with dimensions close to the lower bound. Then, they use explicit values of {\chi(g)} for {\chi} a Weil character to show that their lemma works when the size {|C_G(g)|} of the {G}-conjugacy class of {g} is small.

In particular, the proof of Ore’s conjecture will be complete if one can show that an element {g} with large conjugacy class {C_G(g)} is a commutator. In this direction, LOST introduce the concept of breakable elements and they show by induction that breakable elements are commutators. For example, they prove that:

Theorem 11 Let {g\in SL_n(\mathbb{F}_q)} be a breakable element, that is, {g\in SL_{m}(\mathbb{F}_q)\times SL_{n-m}(\mathbb{F}_q)} with {2\leq m\leq n-m}. Then, {g} is a commutator.

Here, we said that the proof goes by induction because one can assume that this statement is true for {SL_{m}(\mathbb{F}_q)} and {SL_{n-m}(\mathbb{F}_q)} in order to show the statement for {SL_{n}(\mathbb{F}_q)}. However, one has to be careful here because some small cases such as {SL_2(\mathbb{F}_3)} require special attention.

Finally, it remains to deal with unbreakable elements with large conjugacy classes. As it is shown by LOST, an unbreakable element that is not covered in their “Cauchy-Schwarz inequality” argument has a really large conjugacy class: for instance, for {G=Sp_{2n}(\mathbb{F}_2)}, they show (sometimes by computing characters) that if {g} is unbreakable, then {|C_G(g)|<2^{2n+15}} is small as far as their arguments are concerned. Thus, there is a rather small number of such large conjugacy classes of unbreakable elements, so that one can hope to show that the elements in such a small list of conjugacy classes are commutators by simply exhibiting random commutators belonging to each of them, and this is precisely what LOST do to complete the proof of their theorem.

After presenting this beautiful sketch of proof of Ore’s conjecture/LOST theorem, G. Malle decided to comment on the relationship between Ore’s conjecture and word maps as the last topic of his talk.

4. Word maps

Let {F_r} denote the free group on {r} generators {x_1,\dots, x_r} and let {G} be any group. Given a word {w\in F_r}, {w=x_{i_1}\dots x_{i_n}}, we define the word map {f_{w,G}:G^r\rightarrow G} as

\displaystyle f_{w,G}(g_1,\dots, g_r)=g_{i_1}\dots g_{i_n}

We will denote the image of the word map {f_{w,G}} by {w(G)}.

Example 1 For the word {w=[x_1,x_2]}, the surjectivity of the word map {f_{w,G}} is equivalent to Ore’s conjecture for {G}.

In 1983, A. Borel showed that the following “almost surjectivity” result for word maps on algebraic groups:

Theorem 12 (Borel (1983)) Let {G} be a semisimple linear algebraic group over an algebraically closed field. Then, for any non-trivial word {1\neq w\in F_r}, the word map {f_{w,G}} is dominant (i.e., its image {w(G)} contains a Zariski open and dense subset of {G}).

Proof: Firstly, by taking the universal cover of {G}, one can see that it suffices to prove the result when {G} is simple. Secondly, by considering maximal torii, it is possible to further reduce to the case of {G} of type {A} (e.g., if {G=Sp_{2n}}, one reduces the result to the case of {SL_2^n\subset Sp_{2n}}). Finally, one does the type {A} case directly. \Box

Remark 1 Word maps are not surjective in general: for example, in positive characteristic {p}, the image {w(G)} of the word map associated {w=x^p} does not contain regular unipotent elements.

This theorem allows to deduce the following result of M. Larsen, A. Shalev and P. Tiep:

Theorem 13 (Larsen-Shalev-Tiep (2011)) Let {1\neq w_1, w_2\in F_r} be two non-trivial words. Then, there exists an integer {N=N(w_1,w_2)} such that {w_1(G)w_2(G)=G} for all finite non-Abelian simple groups {G} with {|G|\geq N}.

As a corollary of this result, we have:

Corollary 14 If {1\neq w\in F_r}, then there exists {N=N(w)} such that {w(G)^2=G} for all (finite non-Abelian simple {G} with) {|G|\geq N}. In particular, {|w(G)|\geq\sqrt{|G|}}.

A short sketch of proof of Theorem 13 goes as follows. We separate the discussion into two cases: groups of bounded rank and groups of unbounded rank.

For the bounded rank case, Larsen-Shalev-Tiep show that:

Theorem 15 (Larsen-Shalev-Tiep) Let {1\neq w\in F_r} and let {\{G(q)\}} be an infinite family of finite simple groups of fixed Lie type (e.g., {PSL_n(\mathbb{F}_q)} with {n} fixed). Then, there exists {q_0} such that, for each {q\geq q_0}, the set {w(G(q))} contains regular elements from any maximal torus of {G(q)}.

Then, they conclude the bounded rank case essentially by combining this theorem with the result of Ellers-Gordeev in Corollary 5 above.

Finally, for groups with unbounded rank, they use that for all finite simple groups of Lie type except type {D_{2n}}, there are pairs {T_1} and {T_2} of maximal torii such that if {\chi(s)\chi(t)\neq 0} for some {\chi\in Irr(G)}, then {\chi} is the trivial character {1_G} or the Steinberg character {St_G}. At this point, G. Malle ran out of time and thus he decided to finish his talk here.

5. Epilogue

Closing this post, let me mention three question posed to G. Malle after the end of his talk:

  • Y. de Cornulier asked about estimates on the number of ways of writing {g\in G} as a commutator. Here, G. Malle mentioned that it is easy to produce lower bounds (essentially by changing the conjugacy classes of the elements you use to write {g}), but he was not sure about upper bounds.
  • J.-P. Serre asked about the nature of computer part of LOST’s proof (i.e., if there were “inner checks” for correctness, etc.) and G. Malle assured him that the computer programs used are not extremely sophisticated (they are based on GAP and Magma) and thus it is unlikely that there are errors coming from the corresponding codes. (Apparently J.-P. Serre was not completely satisfied with this answer as I could deduce from his facial expression…)
  • I asked about whether one can write {g\in G} as the commutator of a pair of elements {x,y} generating {G} (here I had in mind some potential links to origamis/square-tiled surfaces…), and G. Malle told me that, at least for certain families, this is likely to be known (and it is probably contained in the most recent LOST papers), but he was not aware of a reference were the general problem is treated.

Responses

  1. The answer to your last question is “no”. I posed the same question some time ago to Shalev, and he said that it was shown already by BH Neumann,
    e.g., even A_5 is a counter example and there are many others.

    • Thanks for your answer!

      In fact, after writing this post, I took a look in the literature, and there are several papers on the question I posed and also on the related problem of understanding T_2-systems in certain families of groups, e.g., as you pointed out, BH Newman, Higuchi and Miyamoto studied this for alternating groups, and, more recently, Evans, Guralnick, Pak, McCullough and Wanderley studied this for PSL_2(\mathbb{F}_{p^s}).

      Furthermore, it seems that the presence of several T_2-systems for certain groups has some interesting consequences in the application I had in mind (i.e., Lyapunov spectra of square-tiled surfaces [a.k.a. origamis]) and I hope to write more on this in the future…

      • There is a weaker question for which I do not know any answer.
        Trying to make the subgroup in the representation of $g=[x,y]$
        as large as possible, can one, say, bound its index
        uniformly within some family of groups of Lie type
        (fixing either q, or rank) ?


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