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?



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


