All polytopes are coset geometries: characterizing automorphism groups of k-orbit abstract polytopes

Abstract polytopes generalize the classical notion of convex polytopes to more general combinatorial structures. The most studied ones are regular and chiral polytopes, as it is well-known, they can be constructed as coset geometries from their automorphism groups. This is also known to be true for 2- and 3- orbit 3-polytopes. In this paper we show that every abstract $n$-polytope can be constructed as a coset geometry. This construction is done by giving a characterization, in terms of generators, relations and intersection conditions, of the automorphism group of a $k$-orbit polytope with given symmetry type graph. Furthermore, we use these results to show that for all $k\neq 2$, there exist $k$-orbit $n$-polytopes with Boolean automorphism groups, for all $n\geq 3$.Comment: 41 pages, 6 figure

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 \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 $n\geq 4$

A characterization of triangulations of closed surfaces

In this paper we prove that a finite triangulation of a connected closed surface is completely determined by its intersection matrix. The \emph{intersection matrix} of a finite triangulation, $K$, is defined as $M_{K}=(dim(s_{i}\cap s_{j}))_{0\leq i,0\leq j}^{n-1}$, where $K_{2}=\{s_{0}, \ldots s_{n-1}\}$ is a labelling of the triangles of $K$.Comment: Submitted to EUROCOMB 201