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A lattice $H$ in a locally compact group $G$ is a discrete subgroup such that the coset space $G/H$ admits a finite $G$-invariant measure.

I have read several places that any lattice H in $\operatorname{SL}_n(\mathbb{R})$ which is contained in $\operatorname{SL}_n(\mathbb{Z})$ must have finite index in $\operatorname{SL}_n(\mathbb{Z})$. But I have been unable to prove this. I have tried using the correspondence between the Haar measure on $\operatorname{SL}_n(\mathbb{R})$ and the counting measure on $\operatorname{SL}_n(\mathbb{Z})$, where we can partition $\operatorname{SL}_n(\mathbb{R})$ into sets each containing one element of $\operatorname{SL}_n(\mathbb{Z})$, and then normalizing s.t. each of these has measure one. But this seemed to lead nowhere. Also just restricting the measure on $\operatorname{SL}_n(\mathbb{R})/H$ to $\operatorname{SL}_n(\mathbb{Z})/H$ does not work either since the latter has measure zero.

Thanks a lot to the ones who will answer.

Samuel Adrian Antz
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user275409
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  • A much better proof than in the two answers is using Riemannian geometry, by working with left-invariant Riemannian metrics and corresponding volumes of the quotient-orbifolds/manifolds. – Moishe Kohan Jul 16 '22 at 15:47

2 Answers2

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First, for $H\subset \Gamma\subset G$ with $G$ unimodular, $\Gamma$ discrete, fixing a Haar measure on $G$, there is a unique $G$-invariant measure on $\Gamma\backslash G$ such that $$ \int_{\Gamma\backslash G} \sum_{\gamma\in \Gamma} \varphi(\gamma\cdot g)\;dg \;=;\ \int_G \varphi(g)\;dg $$ for all $\varphi\in C^o_c(G)$. Suppose $\Gamma\backslash G$ has finite measure. Similarly, by the same general uniqueness results, there is a unique measure on $H\backslash G$ such that $$ \int_{\Gamma\backslash G} \sum_{\gamma\in H\backslash \Gamma} \varphi(hg)\;dg \;=\; \int_{H\backslash G} \varphi(g)\;dg $$ for all $\varphi\in C^o_c(H\backslash G)$. This set-up answers most questions about the trio $H\subset \Gamma \subset G$. For example, yes, if $H\backslash G$ has finite volume, then $H$ must be of finite index in $\Gamma$, or else the sum over $H\backslash \Gamma$ is infinite...

paul garrett
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  • Thanks for your answer! Do you know some references, where i can find the results about the existence of such measures? – user275409 Dec 31 '15 at 08:13
  • Existence of such measures would follow from Riesz-Markov-Kakutani, together with surjectivity of the obvious averaging maps. Probably this is proven many places, but hidden away, ... Perhaps Weil's book on integration on topological groups, or Pontryagin's, would be historically first. Many peoples' on-line notes include such points, e.g., my http://www.math.umn.edu/~garrett/m/v/unitary_of_top.pdf in section 4 discusses averaging maps. – paul garrett Dec 31 '15 at 18:31
  • Reiter's book is good. – Mr. Cooperman Jan 04 '16 at 01:27
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If $F\subset\operatorname{SL}_n(\mathbb{R})$ is a (measurable) fundamental domain for the left-action of $\operatorname{SL}_n(\mathbb{Z})$ on $\operatorname{SL}_n(\mathbb{R})$, and $\{k_i\}_{i\in H\backslash\operatorname{SL}_n(\mathbb{Z})}$ is a collection of representatives from the cosets of $H$ in $\operatorname{SL}_n(\mathbb{Z})$, then $$\cup_ik_iF$$ is a fundamental domain for the action of $H$ on $\operatorname{SL}_n(\mathbb{R})$. There must therefore be only finitely many $k_i$ since $H$ is a lattice.

Samuel Adrian Antz
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Mr. Cooperman
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