Let $G$ be a semisimple Lie group with finite center, $K$ a maximal compact subgroup, and $d=\dim(G/K)$. Let $\Gamma$ be a non-cocompact discrete subgroup of $G$.

Is it true that the virtual cohomological dimension (vcd) of $\Gamma$ is $<d$?

The virtual cohomological dimension in this case is the cohomological dimension of some/every torsion-free finite-index subgroup. I guess it's also the rational cohomological dimension.

Remarks:

• if $\Gamma$ is cocompact the vcd equals $d$;
• in general, the vcd is $\le d$;
• if $\Gamma$ is a non-cocompact lattice, then the vcd is $<d$ (at least in the arithmetic case, where it's related to the $\mathbf{Q}$-rank, cf work of Borel-Serre, and also in the rank-1 case; I think the general case follows).

I'd also be interested by variants of this question, where vcd is replaced by the asymptotic dimension, or by Roe's coarse cohomological dimension.

Let $G$ be a semisimple Lie group with finite center, $K$ a maximal compact subgroup, and $d=\dim(G/K)$. Let $\Gamma$ be a non-cocompact discrete subgroup of $G$. [Edit: assume that $\Gamma$ is virtually torsion-free (which is automatic if $G$ is linear and $\Gamma$ is finitely generated).]

Is it true that the virtual cohomological dimension (vcd) of $\Gamma$ is $<d$?

The virtual cohomological dimension in this case is the cohomological dimension of some/every torsion-free finite-index subgroup. I guess it's also the rational cohomological dimension.

Remarks:

• if $\Gamma$ is cocompact the vcd equals $d$;
• in general, the vcd is $\le d$;
• if $\Gamma$ is a non-cocompact lattice, then the vcd is $<d$ (at least in the arithmetic case, where it's related to the $\mathbf{Q}$-rank, cf work of Borel-Serre, and also in the rank-1 case; I think the general case follows).

I'd also be interested by variants of this question, where vcd is replaced by the asymptotic dimension, or by Roe's coarse cohomological dimension.