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 $\C$ and a subgroup $N \leq \C$. Subgroups $N \leq \C$ that give rise to abstract polytopes through such construction are called {\em sparse}. If, further, the stabilizer of a base flag of the poset is precisely $N$, then $N$ is said to be {\em semisparse}. In \cite[Conjecture 5.2]{hartley1999more} Hartley 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 $n\geq 4$.