A note on the rational homological dimension of lattices in positive characteristic

Let $k$ be the function field of an irreducible projective smooth curve $C$ defined over a finite field $\mathbb{F}_{q}$. Let $S$ be a finite non-empty set of (closed) points of $C$. Let ${\mathcal{O}}_{S}$ be the ring of rational functions whose poles lie in $S$. For each $p\in S$ there is a discrete valuation $\nu_{x}$ of $k$ such that $\nu_{p}(f)$ is the order of vanishing of $f$ at $p$. The valuation ring ${\mathcal{O}}_{p}$ is the ring of functions that do not have a pole at $p$, that is

Let $\bar{k}$ denote the algebraic closure of $k$. Let $\mathbf{G}$ be an affine group scheme defined over $\bar{k}$ such that $\mathbf{G}(\bar{k})$ is almost simple. For each $p\in S$ there is a completion $k_{p}$ of $k$ and the group $\mathbf{G}(k_{p})$ acts on the Bruhat–Tits building $X_{p}$. Thus, we may embed $\mathbf{G}({\mathcal{O}}_{S})$ diagonally into the product $\prod_{p\in S}\mathbf{G}(k_{p})$ as an arithmetic lattice.

The rational cohomological dimension of a group $\Gamma$ is defined to be

the rational homological dimension is defined completely analogously as

In [Gan12] it is shown that $\mathrm{cd}_{\mathbb{Q}}(\mathbf{G}({\mathcal{O}}_{S}))=\prod_{p\in S}\dim X_{p}$. In light of this Ian Leary asked the author what is $\mathrm{hd}_{\mathbb{Q}}(\mathbf{G}({\mathcal{O}}_{S}))$?

More generally, we obtain the following.

The author expects these results are well-known, however, they do not appear in the literature so we take the opportunity to record them here.

Let $\Gamma$ be a group. Both $\Gamma$ and the complex group algebra $\mathbb{C}\Gamma$ act by left multiplication on the Hilbert space $\ell^{2}\Gamma$ of square-summable sequences. The group von Neumann algebra ${\mathcal{N}}\Gamma$ is the ring of $\Gamma$-equivariant bounded operators on $\ell^{2}G$. The non-zero divisors of ${\mathcal{N}}G$ form an Ore set and the Ore localization of ${\mathcal{N}}\Gamma$ can be identified with the ring of affiliated operators ${\mathcal{U}}\Gamma$.

There are inclusions $\mathbb{Q}\Gamma\subseteq{\mathcal{N}}\Gamma\subseteq\ell^{2}\Gamma\subseteq{\mathcal{U}}\Gamma$ and it is also known that ${\mathcal{U}}\Gamma$ is a self-injective ring which is flat over ${\mathcal{N}}\Gamma$. For more details concerning these constructions we refer the reader to [Lüc02] and especially to Theorem 8.22 of Section 8.2.3 therein. The von Neumann dimension and the basic properties we need can be found in [Lüc02, Section 8.3].

The $\ell^{2}$-Betti numbers of a group $\Gamma$, denoted $b_{i}^{(2)}(\Gamma)$, are then defined to be the von-Neumann dimensions of the homology groups $H_{i}(\Gamma;{\mathcal{U}}\Gamma)$. The following lemma is a triviality.

Two countable groups $\Gamma$ and $\Lambda$ are said to be measure equivalent if there exist commuting, measure-preserving, free actions of $\Gamma$ and $\Lambda$ on some infinite Lebesgue measure space $(\Omega,m)$, such that the action of each of the groups $\Gamma$ and $\Lambda$ admits a finite measure fundamental domain. The key examples of measure equivalent groups are lattices in the same locally-compact group [Gro93]. The relevance of this for us is the following deep theorem of Gaboriau.