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.



"Dimension" of discrete subgroups of infinite covolume in Lie groups
First of all, since you are not assuming finite generation, you should at least assume that $\Gamma$ is virtually torsion-free. (Otherwise, you need to work rationally and reprove Whitehead's lemma, in the setting of orbifolds/orbicomplexes; see below.) 


With this extra assumption, here is a proof:


WLOG, $\Gamma$ is torsion-free; let $X=G/K$ be the associated symmetric space so $M=\Gamma\backslash X$ is an open $d$-dimensional manifold. Whitehead proved in Lemma 2.1 in 


"The immersion of an open 3-manifold in euclidean space", Proc. London Math. Soc 11 1961, 81-90.


that every open triangulated $d$-dimensional manifold $M$ deformation retracts to its $d-1$-dimensional skeleton $Y\subset M$.


See also Andy Putman's write-up of the proof [here](https://www3.nd.edu/~andyp/notes/NoncompactSurfaceFree.pdf). 


Thus, in your case, there exists a $d-1$-dimensional $Y=K(\Gamma,1)$. Hence, $cd(\Gamma)\le d-1$. 


Edit: If $X$ is an irreducible symmetric space of rank $\ge 2$, I do not know of any examples of discrete subgroups $\Gamma< G$ (which are not non-uniform lattices) of vcd equal to $d-1$. (Most likely, such subgroups simply do not exist.) In contrast, such examples abound in the rank 1 case (more precisely, for real and complex hyperbolic spaces). 


