Are there any known examples of groups that virtually split that don't have a codimension-1 subgroup?



Are there any known examples of groups that virtually split that don't have a codimension-1 subgroup?
No. For a finitely generated group, the property "have no codimension 1 subgroup" is called Property FW. Property FW passes to finite index subgroups, so excludes virtual splittings. 


Actually for an infinitely generated group, the existence of a "codimension 1 subgroup", the transitive version of Property FW, is still discarded by virtual splittings, but there are maybe a few details to check. I'll assume in the absence of context that you're interested by finitely generated groups.


Edit: here are missing references. That Property FW passes to finite index subgroups is Proposition 5.B.1 in [this arxiv paper](https://arxiv.org/abs/1302.5982). The non-existence of a "codimension 1 subgroup" is called Property FW' (and equivalent to Property FW for finitely generated groups, but not in general). That is passes to finite index subgroups is obtained by a similar argument. The whole point is that when one induces an action of a finite index subgroup to the whole group (see §4.F in the above reference), this maps a transitive action to an action with finitely many orbits. I don't use CAT(0) cube complexes for this.


