Measure on a quotient

Question

Can anyone explain me the following :

let $M$ be a hyperbolic manifold and $\Gamma = \Pi_1(M) \subset Iso(\mathbb{H}^n) $. How does the Haar measure on $Iso(\mathbb{H}^n) $ induces a measure on $Iso(\mathbb{H}^n)/ \Gamma $ ?

This statement can be found in Thurston's notes on $3$-manifolds, in chapter 6, to define Gromov norm

Thank you

Answer 1

The process by which Haar measure (sometimes) descends to a quotient consists of: (i) restricting the measure to a fundamental domain; (ii) pushing it forward under the quotient map. This requires two ingredients:

1. a fundamental domain of finite measure (which we have here, since the action is cocompact)
2. Haar measure should be both left- and right- invariant (also the case here, and is mentioned by Thurston)

In some situations one gets 2 from 1, as in Proposition 9.20 in this book.

Answer 2

In addition to the viewpoint of pushing the measure down via a fundamental domain, and the concommitant worry about dependence on that choice, or worry about finding a fundamental domain, or worrying about a-priori finite measure... one may also specify a measure on $\Gamma\backslash G$ for unimodular $G$ and unimodular $\Gamma$ (closed subgroup, discrete or not!), by requiring $$ \int_{\Gamma\backslash G}\Big(\int_\Gamma f(\gamma\cdot \dot{g})\,d\gamma\Big)d\dot{g} $$ It is not hard to show that the averaging map $f(g)=\int_\Gamma f(\gamma\cdot g)d\gamma$ is a surjection from compactly-supported continuous functions on $G$ to compactly-supported continuous functions on the quotient, so the Riesz-Kakutani-Markov theorem gives a unique measure.