Let $G$ be a Lie group and $\Gamma$ be a discrete subgroup of $G$. Definition 1.3.5 of Dave Witte Morris's Introduction to Arithmetic Groups says the following:
We say that $\Gamma$ is a lattice in $G$ if $\Gamma\setminus G$ has finite volume (with respect to the Haar measure on $G$).
I am unable to understand this definition. What is meant by saying "$\Gamma\setminus G$ has finite volume with respect to the Haar measure on $G$?"
Best Answer
A Haar measure on $G$ induces a canonical measure on the coset space $\Gamma\backslash G$. There are various ways of defining it, but here's probably the simplest: a Haar measure on $G$ can be represented by a left invariant volume form $\omega$, and then you can push that volume form down to $\Gamma\backslash G$ along the quotient map (which is a local diffeomorphism). The pushforward is well-defined since $\omega$ is left invariant, and in particular invariant under left multiplication by any element of $\Gamma$.