3 clarified the question

Let $G$ be a locally compact group and $\Gamma$ a lattice (=discrete subgroup of $G$ such that $G/\Gamma$ carries a probability measure $\mu$ that is invariant under the action of $G$ by left-multiplication).

My vague question is: "How to measure the lack of cocompactness of $\Gamma$"?.

Edit: My question was indeed unclear. I am not looking for a criterion that says me whether a lattice is cocompact. Given a non-cocompact lattice, I am looking for a way of quantifying "how much non-cocompact" it is. I propose below such a measurement (and ask what can be said about it), but if somebody has other propositions I will be happy.

A natural such measurement in the case when $G$ has a compact generating set is the following. One can equip $G$ with the word length metric $d$ corresponding to some compact generating set, and consider the sequence $n \mapsto u_n=1 - \mu (B(0,n) \Gamma)$. It is easy to see that $\Gamma$ is cocompact if and only if $u_n=0$ for all sufficiently large $n$. The speed of convergence of $u_n$ to zero in some sense should measure how far we are from this ideal situation. This is related to the integrability properties of $w \mapsto d(1,w)$ on a suitably chosen fundamental domain $\Omega \in G$ for the action of $\Gamma$ on $G$ by right-multiplication.

Has this been studied? What is the typical behaviour of $u_n$ in the case of (arithmetic) lattices in Lie or algebraic groups? In a first time I would already be happy to have an answer for $\Gamma=SL(3,\mathbf Z)$. My guess is that number theorists may have already studied this.

2 Removed last sentence

Let $G$ be a locally compact group and $\Gamma$ a lattice (=discrete subgroup of $G$ such that $G/\Gamma$ carries a probability measure $\mu$ that is invariant under the action of $G$ by left-multiplication).

My vague question is: "How to measure the lack of cocompactness of $\Gamma$"?.

A natural such measurement in the case when $G$ has a compact generating set is the following. One can equip $G$ with the word length metric $d$ corresponding to some compact generating set, and consider the sequence $n \mapsto u_n=1 - \mu (B(0,n) \Gamma)$. It is easy to see that $\Gamma$ is cocompact if and only if $u_n=0$ for all sufficiently large $n$. The speed of convergence of $u_n$ to zero in some sense should measure how far we are from this ideal situation. This is related to the integrability properties of $w \mapsto d(1,w)$ on a suitably chosen fundamental domain $\Omega \in G$ for the action of $\Gamma$ on $G$ by right-multiplication.

Has this been studied? What is the typical behaviour of $u_n$ in the case of (arithmetic) lattices in Lie or algebraic groups? In a first time I would already be happy to have an answer for $\Gamma=SL(3,\mathbf Z)$. My guess is that number theorists may have already studied this, but I am not at all familiar with that subject.

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# Measuring how far from being cocompact a lattice is

Let $G$ be a locally compact group and $\Gamma$ a lattice (=discrete subgroup of $G$ such that $G/\Gamma$ carries a probability measure $\mu$ that is invariant under the action of $G$ by left-multiplication).

My vague question is: "How to measure the lack of cocompactness of $\Gamma$"?.

A natural such measurement in the case when $G$ has a compact generating set is the following. One can equip $G$ with the word length metric $d$ corresponding to some compact generating set, and consider the sequence $n \mapsto u_n=1 - \mu (B(0,n) \Gamma)$. It is easy to see that $\Gamma$ is cocompact if and only if $u_n=0$ for all sufficiently large $n$. The speed of convergence of $u_n$ to zero in some sense should measure how far we are from this ideal situation. This is related to the integrability properties of $w \mapsto d(1,w)$ on a suitably chosen fundamental domain $\Omega \in G$ for the action of $\Gamma$ on $G$ by right-multiplication.

Has this been studied? What is the typical behaviour of $u_n$ in the case of (arithmetic) lattices in Lie or algebraic groups? In a first time I would already be happy to have an answer for $\Gamma=SL(3,\mathbf Z)$. My guess is that number theorists may have already studied this, but I am not at all familiar with that subject.