Sparse groups need not be semisparse

TL;DR

This paper disproves Hartley's conjecture that all sparse groups are semisparse by providing counterexamples in ranks four and higher, using non-faithful maniplexes to construct polytopes.

Contribution

The paper demonstrates that sparse groups are not necessarily semisparse, refuting Hartley's conjecture for ranks four and above through explicit examples.

Findings

Sparse groups can be non-semisparse.

Hartley's conjecture holds for rank 3.

Counterexamples exist for ranks ≥4.

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 $n\geq 4$.