31 research outputs found
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
-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 -orbit polytope with
given symmetry type graph. Furthermore, we use these results to show that for
all , there exist -orbit -polytopes with Boolean automorphism
groups, for all .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 , then 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
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, , is defined as
, where is a labelling of the triangles of .Comment: Submitted to EUROCOMB 201