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Suppose a Fuchsian group $\Gamma$ is derived from a division quaternion algebra. Then the quotient space $\Gamma\backslash \mathcal{H}$ is compact.

I am reading the book "Fuchsian Groups" of Svetlana Katok. In Theorem 5.4.1 (as above), there is only a proof for the simplest case when $A$ is a division algebra over $\mathbb{Q}$: $ A=\left ( \frac{a,b}{\mathbb{Q}} \right ), a>0, \mathcal{O}=\left \{ x\in A | x_0, x_1, x_2 , x_{3}\in \mathbb{Z} \right \}$

My question:

  1. Is there any idea or a proof in the general case somewhere?
  2. Are there some special properties of mixing flows on that surface which is wrong on compact hyperbolic surfaces in general?

Hoping someone can help me. Thanks in advance !

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  • $\begingroup$ A general proof that an arithmetic Fuchsian group derived from a quaternion division algebra is cocompact appears in Maclachlan and Reid's book (The Arithmetic of Hyperbolic 3-Manifolds). It is Theorem 8.12. In fact, this proof applies to discrete subgroups of products of SL(2,R) and SL(2,C) arising from quaternion algebras as well. Thus, for instance, their theorem implies your question for arithmetic Kleinian groups. $\endgroup$
    – user1073
    Commented Apr 9, 2017 at 13:42

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1 - There is an explicit reference given in the book: Borel, Harish-chandra, arithmetic subgroups of algebraic groups, 1962. This is the general result for matrix groups. A simpler proof has been given by Mostow and Tamagawa (1962). It is based on the lemmas by Mahler and Minkowski on the manifold of lattices, as in the book of S. Katok.

2 - There is nothing special about arithmetic groups. The geodesic flow is mixing with respect to the Haar measure on all quotients of finite volume. This is due to Hedlund (1939), many books on ergodic theory contain short proofs of this result, e.g. Nicholls, ergodic theory of discrete groups, Cornfeld, Fomin, Sinai, ergodic theory etc. Mixing actually holds for a quite large class of hyperbolic dynamical systems, e.g. Anosov flows, Teichmuller flows.

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  • $\begingroup$ Actually, it is quite likely that the speed of mixing on arithmetic groups is higher than it is on arbitrary groups - I am guessing it is related to Selberg's eigenvalue conjecture. $\endgroup$
    – Igor Rivin
    Commented Apr 9, 2017 at 21:05

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