Sparse groups need not be semisparse

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AbstractIn 1999 Michael Hartley showed that any abstract polytope can be constructed as a double coset poset, by means of a C-group $$\mathcal {W}$$ W and a subgroup $$N \le \mathcal {W}$$ N ≤ W . Subgroups $$N \le \mathcal {W}$$ N ≤ 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 $$n\ge 4$$ n ≥ 4 .