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Over the past few months or so, I have been brushing up on my Lie group theory, as part of my project to fully understand the theory surrounding Hilbert’s fifth problem. Every so often, I encounter a basic fact in Lie theory which requires a slightly non-trivial “trick” to prove; I am recording two of them here, so that I can find these tricks again when I need to.
The first fact concerns the exponential map from a Lie algebra of a Lie group to that group. (For this discussion we will only consider finite-dimensional Lie groups and Lie algebras over the reals .) A basic fact in the subject is that the exponential map is locally a homeomorphism: there is a neighbourhood of the origin in that is mapped homeomorphically by the exponential map to a neighbourhood of the identity in . This local homeomorphism property is the foundation of an important dictionary between Lie groups and Lie algebras.
It is natural to ask whether the exponential map is globally a homeomorphism, and not just locally: in particular, whether the exponential map remains both injective and surjective. For instance, this is the case for connected, simply connected, nilpotent Lie groups (as can be seen from the Baker-Campbell-Hausdorff formula.)
The circle group , which has as its Lie algebra, already shows that global injectivity fails for any group that contains a circle subgroup, which is a huge class of examples (including, for instance, the positive dimensional compact Lie groups, or non-simply-connected Lie groups). Surjectivity also obviously fails for disconnected groups, since the Lie algebra is necessarily connected, and so the image under the exponential map must be connected also. However, even for connected Lie groups, surjectivity can fail. To see this, first observe that if the exponential map was surjective, then every group element has a square root (i.e. an element with ), since has as a square root for any . However, there exist elements in connected Lie groups without square roots. A simple example is provided by the matrix
in the connected Lie group . This matrix has eigenvalues , . Thus, if is a square root of , we see (from the Jordan normal form) that it must have at least one eigenvalue in , and at least one eigenvalue in . On the other hand, as has real coefficients, the complex eigenvalues must come in conjugate pairs . Since can only have at most eigenvalues, we obtain a contradiction.
However, there is an important case where surjectivity is recovered:
Proposition 1 If is a compact connected Lie group, then the exponential map is surjective.
Proof: The idea here is to relate the exponential map in Lie theory to the exponential map in Riemannian geometry. We first observe that every compact Lie group can be given the structure of a Riemannian manifold with a bi-invariant metric. This can be seen in one of two ways. Firstly, one can put an arbitrary positive definite inner product on and average it against the adjoint action of using Haar probability measure (which is available since is compact); this gives an ad-invariant positive-definite inner product on that one can then translate by either left or right translation to give a bi-invariant Riemannian structure on . Alternatively, one can use the Peter-Weyl theorem to embed in a unitary group , at which point one can induce a bi-invariant metric on from the one on the space of complex matrices.
As is connected and compact and thus complete, we can apply the Hopf-Rinow theorem and conclude that any two points are connected by at least one geodesic, so that the Riemannian exponential map from to formed by following geodesics from the origin is surjective. But one can check that the Lie exponential map and Riemannian exponential map agree; for instance, this can be seen by noting that the group structure naturally defines a connection on the tangent bundle which is both torsion-free and preserves the bi-invariant metric, and must therefore agree with the Levi-Civita metric. (Alternatively, one can embed into a unitary group and observe that is totally geodesic inside , because the geodesics in can be described explicitly in terms of one-parameter subgroups.) The claim follows.
Remark 1 While it is quite nice to see Riemannian geometry come in to prove this proposition, I am curious to know if there is any other proof of surjectivity for compact connected Lie groups that does not require explicit introduction of Riemannian geometry concepts.
The other basic fact I learned recently concerns the algebraic nature of Lie groups and Lie algebras. An important family of examples of Lie groups are the algebraic groups – algebraic varieties with a group law given by algebraic maps. Given that one can always automatically upgrade the smooth structure on a Lie group to analytic structure (by using the Baker-Campbell-Hausdorff formula), it is natural to ask whether one can upgrade the structure further to an algebraic structure. Unfortunately, this is not always the case. A prototypical example of this is given by the one-parameter subgroup
of . This is a Lie group for any exponent , but if is irrational, then the curve that traces out is not an algebraic subset of (as one can see by playing around with Puiseux series).
of . This is a Lie group for any exponent , but if is irrational, then the curve that traces out is not an algebraic subset of (as one can see by playing around with Puiseux series).
This is not a true counterexample to the claim that every Lie group can be given the structure of an algebraic group, because one can give a different algebraic structure than one inherited from the ambient group . Indeed, is clearly isomorphic to the additive group , which is of course an algebraic group. However, a modification of the above construction works:
Proposition 2 There exists a Lie group that cannot be given the structure of an algebraic group.
of , with an irrational number. This is a three-dimensional (metabelian) Lie group, whose Lie algebra is spanned by the elements
with the Lie bracket given by
As such, we see that if we use the basis to identify to , then adjoint representation of is the identity map.
If is an algebraic group, it is easy to see that the adjoint representation is also algebraic, and so is algebraic in . Specialising to our specific example, in which adjoint representation is the identity, we conclude that if has any algebraic structure, then it must also be an algebraic subgroup of ; but projects to the group (1) which is not algebraic, a contradiction.
A slight modification of the same argument also shows that not every Lie algebra is algebraic, in the sense that it is isomorphic to a Lie algebra of an algebraic group. (However, there are important classes of Lie algebras that are automatically algebraic, such as nilpotent or semisimple Lie algebras.)
Last year, Emmanuel Breuillard, Ben Green, Bob Guralnick, and I wrote a paper entitled “Strongly dense free subgroups of semisimple Lie groups“. The main theorem in that paper asserted that given any semisimple algebraic group over an uncountable algebraically closed field , there existed a free subgroup which was strongly dense in the sense that any non-abelian subgroup of was Zariski dense in . This type of result is useful for establishing expansion in finite simple groups of Lie type, as we will discuss in a subsequent paper.
An essentially equivalent formulation of the main result is that if are two non-commuting elements of the free group on two generators, and is a generic pair of elements in , then and are not contained in any proper closed algebraic subgroup of . Here, “generic” means “outside of at most countably many proper subvarieties”. In most cases, one expects that if are generically drawn from , then will also be generically drawn from , but this is not always the case, which is a key source of difficulty in the paper. For instance, if is conjugate to in , then and must be conjugate in and so the pair lie in a proper subvariety of . It is currently an open question to determine all the pairs of words for which is not generic for generic (or equivalently, the double word map is not dominant).
The main strategy of proof was as follows. It is not difficult to reduce to the case when is simple. Suppose for contradiction that we could find two non-commuting words such that were generically trapped in a proper closed algebraic subgroup. As it turns out, there are only finitely many conjugacy classes of such groups, and so one can assume that were generically trapped in a conjugate of a fixed proper closed algebraic subgroup . One can show that , , and are generically regular semisimple, which implies that is a maximal rank semisimple subgroup. The key step was then to find another proper semisimple subgroup of which was not a degeneration of , by which we mean that there did not exist a pair in the Zariski closure of the products of conjugates of , such that generated a Zariski-dense subgroup of . This is enough to establish the theorem, because we could use an induction hypothesis to find in (and hence in such that generated a Zariski-dense subgroup of , which contradicts the hypothesis that was trapped in for generic (and hence in for all .
To illustrate the concept of a degeneration, take and let be the stabiliser of a non-degenerate -space in . All other stabilisers of non-degenerate -spaces are conjugate to . However, stabilisers of degenerate -spaces are not conjugate to , but are still degenerations of . For instance, the stabiliser of a totally singular -space (which is isomorphic to the affine group on , extended by ) is a degeneration of .
A significant portion of the paper was then devoted to verifying that for each simple algebraic group , and each maximal rank proper semisimple subgroup of , one could find another proper semisimple subgroup which was not a degeneration of ; roughly speaking, this means that is so “different” from that no conjugate of can come close to covering . This required using the standard classification of algebraic groups via Dynkin diagrams, and knowledge of the various semisimple subgroups of these groups and their representations (as we used the latter as obstructions to degeneration, for instance one can show that a reducible representation cannot degenerate to an irreducible one).
During the refereeing process for this paper, we discovered that there was precisely one family of simple algebraic groups for which this strategy did not actually work, namely the group (or the group that is double-covered by this group) in characteristic . This group (which has Dynkin diagram , as discussed in this previous post) has one maximal rank proper semisimple subgroup up to conjugacy, namely , which is the stabiliser of a line in . To find a proper semisimple group that is not a degeneration of this group, we basically need to find a subgroup that does not stabilise any line in . In characteristic larger than three (or characteristic zero), one can proceed by using the action of on the five-dimensional space of homogeneous degree four polynomials on , which preserves a non-degenerate symmetric form (the four-fold tensor power of the area form on ) and thus embeds into ; as no polynomial is fixed by all of , we see that this copy of is not a degeneration of .
Unfortunately, in characteristics two and three, the symmetric form on degenerates, and this embedding is lost. In the characteristic two case, one can proceed by using the characteristic fact that is isomorphic to (because in characteristic two, the space of null vectors is a hyperplane, and the symmetric form becomes symplectic on this hyperplane), and thus has an additional maximal rank proper semisimple subgroup which is not conjugate to the subgroup. But in characteristic three, it turns out that there are no further semisimple subgroups of that are not already contained in a conjugate of the . (This is not a difficulty for larger groups such as or , where there are plenty of other semisimple groups to utilise; it is only this smallish group that has the misfortune of having exactly one maximal rank proper semisimple group to play with, and not enough other semisimples lying around in characteristic three.)
As a consequence of this issue, our argument does not actually work in the case when the characteristic is three and the semisimple group contains a copy of (or ), and we have had to modify our paper to delete this case from our results. We believe that such groups still do contain strongly dense free subgroups, but this appears to be just out of reach of our current method.
One thing that this experience has taught me is that algebraic groups behave somewhat pathologically in low characteristic; in particular, intuition coming from the characteristic zero case can become unreliable in characteristic two or three.
Emmanuel Breuillard, Ben Green, Robert Guralnick, and I have just uploaded to the arxiv our paper “Strongly dense free subgroups of semisimple algebraic groups“, submitted to Israel J. Math.. This paper was originally motivated by (and provides a key technical tool for) another forthcoming paper of ours, on expander Cayley graphs in finite simple groups of Lie type, but also has some independent interest due to connections with other topics, such as the Banach-Tarski paradox.
Recall that one of the basic facts underlying the Banach-Tarski paradox is that the rotation group contains a copy of the free non-abelian group on two generators; thus there exists such that obey no nontrivial word identities. In fact, using basic algebraic geometry, one can then deduce that a generic pair of group elements has this property, where for the purposes of this paper “generic” means “outside of at most countably many algebraic subvarieties of strictly smaller dimension”. (In particular, using Haar measure on , almost every pair has this property.) In fact one has a stronger property, given any non-trivial word , the associated word map from to is a dominant map, which means that its image is Zariski-dense. More succinctly, if is generic, then is generic also.
In contrast, if one were working in a solvable, nilpotent, or abelian group (such as ), then this property would not hold, since every subgroup of a solvable group is still solvable and thus not free (and similarly for nilpotent or abelian groups). (This already goes a long way to explain why the Banach-Tarski paradox holds in three or more dimensions, but not in two or fewer.) On the other hand, a famous result of Borel asserts that for any semisimple Lie group (over an algebraically closed field), and any nontrivial word , the word map is dominant, thus generalising the preceding discussion for . (There is also the even more famous Tits alternative, that asserts that any linear group that is not (virtually) solvable will contain a copy of the free group ; as pointed out to me by Michael Cowling, this already shows that generic pairs of generators will generate a free group, and with a little more effort one can even show that it generates a Zariski-dense free group.)
Now suppose we take two words , and look at the double word map on a semisimple Lie group . If are non-trivial, then Borel’s theorem tells us that each component of this map is dominant, but this does not mean that the entire map is dominant, because there could be constraints between and . For instance, if the two words commute, then must also commute and so the image of the double word map is not Zariski-dense. But there are also non-commuting examples of non-trivial constraints: for instance, if are conjugate, then must also be conjugate, which is also a constraint that obstructs dominance.
It is still not clear exactly what pairs of words have the dominance property. However, we are able to establish that all pairs of non-commuting words have a weaker property than dominance:
Theorem. Let be non-commuting words, and let be generic elements of a semisimple Lie group over an algebraically closed field. Then generate a Zariski-dense subgroup of .
To put it another way, not only contains free subgroups, but contains what we call strongly dense free subgroups: free subgroups such that any two non-commuting elements generate a Zariski-dense subgroup.
Our initial motivation for this theorem is its implications for finite simple groups of Lie type. Roughly speaking, one can use this theorem to show that a generic random walk in such a group cannot be trapped in a (bounded complexity) proper algebraic subgroup of , and this “escape from subgroups” fact is a key ingredient in our forthcoming paper in which we demonstrate that random Cayley graphs in such groups are expander graphs.
It also has implications for results of Banach-Tarski type; it shows that for any semisimple Lie group G, and for generic , one can use to create Banach-Tarski paradoxical decompositions for all homogeneous spaces of . In particular there is one pair of that gives paradoxical decompositions for all homogeneous spaces simultaneously.
Our argument is based on a concept that we call “degeneration”. Let be generic elements of , and suppose for contradiction that generically generated a group whose algebraic closure was conjugate to a proper algebraic subgroup of . Borel’s theorem lets us show that , and latex [w_1(a,b), w_2(a,b)]$ each generate maximal tori of , which by basic algebraic group theory can be used to show that must be a proper semisimple subgroup of of maximal rank. If we were in the model case , then we would already be done, as there are no such maximal rank semisimple subgroups; but in the other groups, such proper maximal semisimple groups unfortunately exist. Fortunately, they have been completely classified, and we take advantage of this classification in our argument.
The degeneration argument comes in as follows. Let be a non-generic pair in . Then lies in the Zariski closure of the generic pairs, which means that lies in the Zariski closure of the set formed by and its conjugates. In particular, if the non-generic pair is such that generates a group that is dense in some proper algebraic subgroup , then is in the Zariski closure of the union of the conjugates of . When this happens, we say that is a degeneration of . (For instance, could be the stabiliser of some non-degenerate quadratic form, and could be the stabiliser of a degenerate limit of that form.)
The key fact we need (that relies on the classification, and a certain amount of representation theory) is:
Proposition. Given any proper semisimple maximal rank subgroup of , there exists another proper semisimple subgroup that is not a degeneration of .
Using an induction hypothesis, we can find pairs such that generate a dense subgroup of , which together with the preceding discussion contradicts the proposition.
The proposition is currently proven by using some known facts about certain representation-theoretic invariants of all the semisimple subgroups of the classical and exceptional simple Lie groups. While the proof is of finite length, it is not particularly elegant, ultimately relying on the numerical value of one or more invariants of being sufficiently different from their counterparts for that one can prevent the latter being a degeneration of the former. Perhaps there is another way to proceed here that is not based so heavily on classification.
Emmanuel Breuillard, Ben Green, and I have just uploaded to the arXiv our paper “Approximate subgroups of linear groups“, submitted to GAFA. This paper contains (the first part) of the results announced previously by us; the second part of these results, concerning expander groups, will appear subsequently. The release of this paper has been coordinated with the release of a parallel paper by Pyber and Szabo (previously announced within an hour(!) of our own announcement).
Our main result describes (with polynomial accuracy) the “sufficiently Zariski dense” approximate subgroups of simple algebraic groups , or more precisely absolutely almost simple algebraic groups over , such as . More precisely, define a -approximate subgroup of a genuine group to be a finite symmetric neighbourhood of the identity (thus and ) such that the product set can be covered by left-translates (and equivalently, right-translates) of .
Let be a field, and let be its algebraic closure. For us, an absolutely almost simple algebraic group over is a linear algebraic group defined over (i.e. an algebraic subvariety of for some with group operations given by regular maps) which is connected (i.e. irreducible), and such that the completion has no proper normal subgroups of positive dimension (i.e. the only normal subgroups are either finite, or are all of . To avoid degeneracies we also require to be non-abelian (i.e. not one-dimensional). These groups can be classified in terms of their associated finite-dimensional simple complex Lie algebra, which of course is determined by its Dynkin diagram, together with a choice of weight lattice (and there are only finitely many such choices once the Lie algebra is fixed). However, the exact classification of these groups is not directly used in our work.
Our first main theorem classifies the approximate subgroups of such a group in the model case when generates the entire group , and is finite; they are either very small or very large.
Theorem 1 (Approximate groups that generate) Let be an absolutely almost simple algebraic group over . If is finite and is a -approximate subgroup of that generates , then either or , where the implied constants depend only on .
The hypothesis that generates cannot be removed completely, since if was a proper subgroup of of size intermediate between that of the trivial group and of , then the conclusion would fail (with ). However, one can relax the hypothesis of generation to that of being sufficiently Zariski-dense in . More precisely, we have
Theorem 2 (Zariski-dense approximate groups) Let be an absolutely almost simple algebraic group over . If is a -approximate group) is not contained in any proper algebraic subgroup of of complexity at most (where is sufficiently large depending on ), then either or , where the implied constants depend only on and is the group generated by .
Here, we say that an algebraic variety has complexity at most if it can be cut out of an ambient affine or projective space of dimension at most by using at most polynomials, each of degree at most . (Note that this is not an intrinsic notion of complexity, but will depend on how one embeds the algebraic variety into an ambient space; but we are assuming that our algebraic group is a linear group and thus comes with such an embedding.)
In the case when , the second option of this theorem cannot occur since is infinite, leading to a satisfactory classification of the Zariski-dense approximate subgroups of almost simple connected algebraic groups over . On the other hand, every approximate subgroup of is Zariski-dense in some algebraic subgroup, which can be then split as an extension of a semisimple algebraic quotient group by a solvable algebraic group (the radical of the Zariski closure). Pursuing this idea (and glossing over some annoying technical issues relating to connectedness), together with the Freiman theory for solvable groups over due to Breuillard and Green, we obtain our third theorem:
Theorem 3 (Freiman’s theorem in ) Let be a -approximate subgroup of . Then there exists a nilpotent -approximate subgroup of size at most , such that is covered by translates of .
This can be compared with Gromov’s celebrated theorem that any finitely generated group of polynomial growth is virtually nilpotent. Indeed, the above theorem easily implies Gromov’s theorem in the case of finitely generated subgroups of .
By fairly standard arguments, the above classification theorems for approximate groups can be used to give bounds on the expansion and diameter of Cayley graphs, for instance one can establish a conjecture of Babai and Seress that connected Cayley graphs on absolutely almost simple groups over a finite field have polylogarithmic diameter at most. Applications to expanders include the result on Suzuki groups mentioned in a previous post; further applications will appear in a forthcoming paper.
Apart from the general structural theory of algebraic groups, and some quantitative analogues of the basic theory of algebraic geometry (which we chose to obtain via ultrafilters, as discussed in this post), we rely on two basic tools. Firstly, we use a version of the pivot argument developed first by Konyagin and Bourgain-Glibichuk-Konyagin in the setting of sum-product estimates, and generalised to more non-commutative settings by Helfgott; this is discussed in this previous post. Secondly, we adapt an argument of Larsen and Pink (which we learned from a paper of Hrushovski) to obtain a sharp bound on the extent to which a sufficiently Zariski-dense approximate groups can concentrate in a (bounded complexity) subvariety; this is discussed at the end of this blog post.