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.
7 comments
Comments feed for this article
29 March, 2011 at 2:44 pm
Anonymous
typo in paragraph 6: “which this” instead of “this this”
[Corrected, thanks – T.]
29 March, 2011 at 4:05 pm
Yiftach
Should be Bob Guralnick rather than Rob (unless there is a mysterious Rob Guralnick in addition to Bob).
[Gah, that was silly of me – corrected, thanks! – T.]
30 March, 2011 at 1:23 pm
Pila
A weakly related question: do you give somewhere in your general advices the way you check your papers (or the way you suggests to check one’s papers) ?
30 March, 2011 at 2:20 pm
Terence Tao
I discuss this sort of thing at
http://www.google.com/buzz/114134834346472219368/WJkEENg19Sz
4 April, 2011 at 3:44 pm
Neville Campbell
Exceptional behavior of small cases is common in group theory. It can be viewed as the reason for the existence of exceptional finite simple groups.
6 April, 2011 at 6:59 pm
Weekly Picks « Mathblogging.org — the Blog
[…] campus interviews at grad schools, punk rock OR explained what OR teaches you about having a baby, Terry Tao posted an exemplary erratum on a paper, Women in mathematics (in Berlin) studied websites of fellow BMS grad students and James Colliander […]
10 September, 2013 at 8:15 am
Expansion in finite simple groups of Lie type | What's new
[…] are also some related results established in the paper. Firstly, as we discovered after writing our first paper, there was one class of algebraic groups for which our demonstration of strongly dense subgroups […]