# Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?

Fix a Lie group $G$ and a discrete subgroup $\Gamma \subset G$. Homogeneous dynamics is about studying the actions of subgroups $H \subset G$ on the quotient $G/\Gamma$.

Does anyone know of an example of a question about $\Gamma$ that was answered by considering these dynamics?

Where to begin!

The ergodicity of non-compact subgroups (singular tori) was used by Margulis to prove that higher rank lattices $$\Gamma$$ are arithmetic.

Once you have that $$\Gamma$$ is arithmetic, this has the following consequences: (1) if $$Comm (\Gamma)$$ is the abstract commensurator, then $$Comm (\Gamma)/\Gamma$$ is infinite. (2) The cohomology groups $$H^*(\Gamma,\mathbb{Z})$$ are finitely generated abelian groups (Raghunathan). (3) The Group $$\Gamma$$ is finitely presented (Borel-Harishchandra).

The Oppenheim conjecture about quadratic forms could be interpreted as a property of the dynamics of $$SO(2,1)$$ action on $$SL(3,\mathbb {R})/SL(3,\mathbb{Z})$$ (and was proved by Margulis).

That normal subgroups of torsion free higher rank arithmetic groups $$\Gamma$$ have finite index is also proved by dynamics of toral actions on $$G/\Gamma$$ (Margulis).

In Zimmer's book, the proof of Borel density theorem (that a lattice is Zariaski dense in $$G$$) is proved using dynamics of $$G$$ action on $$G/\Gamma$$ and also $$G$$ action on projective space.

Yet another Margulis theorem says that higher rank arithmetic groups are not free products (or not even amalgams) ; one part of the proof uses ergodicity of actions of singular tori on $$G/\Gamma$$.

I am sure @YCor knows many more (and also recent) examples.

You may also see https://mathscinet.ams.org/mathscinet-getitem?mr=1898148 for some more examples.

• I'm happy with the result about normal subgroups having finite index, but the other examples are not what I have in mind. Oppenheim is related to the quotient space, but it is not really a statement about the group $\text{SL}(3,\mathbf{Z})$. Similarly, Borel density and arithmeticity are statements about the position of $\Gamma$ as a subgroup inside a larger group $G$. But these do not seem to be intrinsic properties of $\Gamma$ itself. Although the following is unclear to me: is arithmeticity an intrinsic property of a discrete group $\Gamma$? (Zariski density obviously is not)
– Kim
Sep 20 '18 at 18:27
• @Kim technically speaking you're right about arithmeticity, but indeed it allows to prove plenty of things about the underlying group structure (distorsion of elements, description of solvable subgroups, etc)
– YCor
Sep 20 '18 at 19:05
• @Kim I don't know any systematic survey. For distortion of cyclic subgroups, the main article is by Lubotzky-Mozes-Raghunathan.
– YCor
Sep 20 '18 at 19:29
• @Kim: Well, arithmeticity of $\Gamma =G(\mathbb Z)$ can be interpreted (plus Mostow rigidity, which can be proved by using superrigidity (dynamical considerations again) ) as saying that the abstract commensurator of $\Gamma$ contains $\Gamma$ as an infinite index subgroup. Sep 20 '18 at 19:41
• @Kim: I am saying that if $\Gamma$ is a higher rank lattice, then $Comm (\Gamma)/\Gamma$ is infinite. Sep 20 '18 at 19:52

There's a nice proof by Margulis showing that arithmetic subgroups are indeed lattices using the famous Dani-Margulis non-divergence theorem. Actually if you will investigate Ratner's original formulations of her theorems, she usually only assumes that $$\Gamma$$ is discrete, and then concludes something about it having finite-volume.

Furstenberg's proof of Borel's density theorem also come into mind.

Sort of related to dynamics - Kazhdan's property T (i.e. very strong mixing statement) was developed by Kazhdan (and Margulis) to answer Selberg's conjectures about the abelinzation of a lattice.

A more recent development, using the quantitative equidistribution of semisimple periods (proved by Einsiedler-Margulis-Venkatesh), the authors (with the addition of Mohammadi) managed to show ''transport of spectral gap'' of products allowing one to derive a new proof of property tau (for most types of groups, definitely the weird ones included).

Another example can be seen in the proof of Einsiedler-Lindenstrauss-Michel-Venkatesh of Duke's (Linnik-Skubenko) theorem. The statement there is about the $$\Gamma$$-action on $$G/A$$ and then they use duality to transfer it to problem about $$A$$-action over $$\Gamma \backslash G$$. [There's also a whole industry of counting problems and Diophantine approximation problems related to lattice actions a-la Duke-Sarnak, see the recent book by Gorodnik-Nevo.]

I believe the question is exactly what you call ''Homogeneous dynamics''. I guess the subject's founding fathers, back in Jerusalem, will say it is a mix of ergodic theory, representation theory (and so harmonic analysis), Lie and algebraic group theory with some inputs from geometric group theory and number theory (both analytic, geometry of numbers and algebraic nt), rather than just focusing on the ergodic theory of the matters. So I guess the question is what you are willing to assume and what are you after exactly. If you just want direct corollaries of the pointwise ergodic theorem, then it is a whole different story.

• These are all very interesting results, certainly. What I am looking for is the subset of results about intrinsic properties of $\Gamma$ (rather than its properties as a subgroup of $G$).
– Kim
Sep 20 '18 at 19:35
• Some of the proofs of Mostow rigidity for example would use mixing, but I still don't get what you mean by intrinsic, you begin by saying that $\Gamma$ is a lattice in $G$, that gives you quite a bit of information.
– Asaf
Sep 20 '18 at 19:39
• What I mean is that I consider $\Gamma$ as an (isomorphism class of) abstract group. In principle I could embed $\Gamma$ as a lattice in some $G$, as a non-lattice in another $G'$, etc. So being a lattice is not an intrinsic property of $\Gamma$. Intrinsic questions would be things like: What are the normal subgroups of $\Gamma$? Is $\Gamma$ finitely generated? and so on.
– Kim
Sep 20 '18 at 19:43
• What is "the homogeneous dynamics of $G/\Gamma$" without further assumptions on $\Gamma$?
– Asaf
Sep 20 '18 at 20:58
• papers of Benoit and Quint deal with this infinite covolume setting. Sep 21 '18 at 1:43

Kahn and Markovic solved the surface subgroup problem and Ehrenpreis conjecture making use of exponential mixing of the geodesic flow on compact hyperbolic manifolds. The geodesic flow may be thought of as homogeneous dynamics on a diagonal subgroup $$H< G$$ acting on $$G/\Gamma$$, although the proof by Cal Moore only refers to the unit tangent bundle (which is where the geodesic flow lives). At least in the case of $$PSL_2(\mathbb{R})$$ (for hyperbolic surfaces), this is the same thing.

In turn, the surface subgroup problem tells us many more interesting things about $$\Gamma$$ when $$\Gamma$$ is a closed hyperbolic 3-manifold group.

• Could you elaborate? What kind of interesting things do we learn about $\Gamma$ when we know it contains a surface subgroup?
– Kim
Sep 21 '18 at 5:41
• @Kim : this has been generalized by many people to other lattices. Gromov conjectured hyperbolic groups have a surface subgroup. For other consequences, see my survey for the ICM proceedings (you can find it on my homepage). Sep 21 '18 at 14:02
• @Kim, Ian is being modest. Using tools developed by Wise and others, Ian used the work of Kahn—Markovic to deduce that hyperbolic 3-manifold groups are virtually special, which in turn resolved many famous open problems. Sample applications: virtual Haken, virtual fibring, LERF, largeness... All of these can be phrased as group-theoretic properties of $\Gamma$.
– HJRW
Sep 21 '18 at 15:40
• Yes, there’s a lot more that goes into it than just exponential mixing. I guess though that this remains an essential part in this sort of surface subgroup theorem. Sep 21 '18 at 16:48