Sparse groups need not be semisparse

Abstract In 1999 Michael Hartley showed that any abstract polytope can be constructed as a double coset poset, by means of a C-group $$\mathcal {W}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>W</mml:mi> </mml:math> and a subgroup $$N \le \mathcal {W}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>W</mml:mi> </mml:mrow> </mml:math> . Subgroups $$N \le \mathcal {W}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>W</mml:mi> </mml:mrow> </mml:math> 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 $$n\ge 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> .