2 research outputs found
Three-dimensional maps and subgroup growth
In this paper we derive a generating series for the number of cellular
complexes known as pavings or three-dimensional maps, on darts, thus
solving an analogue of Tutte's problem in dimension three.
The generating series we derive also counts free subgroups of index in
via a simple bijection
between pavings and finite index subgroups which can be deduced from the action
of on the cosets of a given subgroup. We then show that this
generating series is non-holonomic. Furthermore, we provide and study the
generating series for isomorphism classes of pavings, which correspond to
conjugacy classes of free subgroups of finite index in .
Computational experiments performed with software designed by the authors
provide some statistics about the topology and combinatorics of pavings on
darts.Comment: 17 pages, 6 figures, 1 table; computational experiments added; a new
set of author
Telescopic groups and symmetries of combinatorial maps
In the present paper, we show that many combinatorial and topological
objects, such as maps, hypermaps, three-dimensional pavings, constellations and
branched coverings of the two--sphere admit any given finite automorphism
group. This enhances the already known results by Frucht, Cori -- Mach\`i,
\v{S}ir\'{a}\v{n} -- \v{S}koviera, and other authors. We also provide a more
universal technique for showing that ``any finite automorphism group is
possible'', that is applicable to wider classes or, in contrast, to more
particular sub-classes of said combinatorial and geometric objects. Finally, we
show that any given finite automorphism group can be realised by sufficiently
many non-isomorphic such entities (super-exponentially many with respect to a
certain combinatorial complexity measure).Comment: 29 pages, 7 figures; final version to appear in Algebraic
Combinatorics https://alco.centre-mersenne.or