5 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
Free subgroups of free products and combinatorial hypermaps
We derive a generating series for the number of free subgroups of finite
index in by using a connection between
free subgroups of and certain hypermaps (also known as ribbon graphs
or "fat" graphs), and show that this generating series is transcendental. We
provide non-linear recurrence relations for the above numbers based on
differential equations that are part of the Riccati hierarchy. We also study
the generating series for conjugacy classes of free subgroups of finite index
in , which correspond to isomorphism classes of hypermaps. Asymptotic
formulas are provided for the numbers of free subgroups of given finite index,
conjugacy classes of such subgroups, or, equivalently, various types of
hypermaps and their isomorphism classes.Comment: 27 pages, 3 figures; supplementary SAGE worksheets available at
http://sashakolpakov.wordpress.com/list-of-papers
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