Sparse groups need not be semisparse

In 1999 Michael Hartley showed that any abstract polytope can be constructed as a double coset poset, by means of a C-group \(\mathcal {W}\) and a subgroup \(N \le \mathcal {W}\). Subgroups \(N \le \mathcal {W}\) that give rise to abstract polytopes through such a construction are called sparse. If, further, the stabilizer of a base flag of the poset is precisely N, then N is said to be semisparse. In [4, Conjecture 5.2] Hartely conjectures that sparse groups are always semisparse. In this paper, we show that this conjecture is in fact false: there exist sparse groups that are not semisparse. In particular, we show that such groups are always obtained from non-faithful maniplexes that give rise to polytopes. Using this, we show that Hartely’s conjecture holds for rank 3, but we construct examples to disprove the conjecture for all ranks \(n\ge 4\).