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 $n$ darts, thus solving an analogue of Tutte's problem in dimension three. The generating series we derive also counts free subgroups of index $n$ in $\Delta^+ = \mathbb{Z}_2*\mathbb{Z}_2*\mathbb{Z}_2$ via a simple bijection between pavings and finite index subgroups which can be deduced from the action of $\Delta^+$ 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 $\Delta^+$. Computational experiments performed with software designed by the authors provide some statistics about the topology and combinatorics of pavings on $n\leq 16$ 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 $\Delta^+ = \mathbb{Z}_p*\mathbb{Z}_q$ by using a connection between free subgroups of $\Delta^+$ 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 $\Delta^+$, 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

Restricted linear congruences

In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence $a_1x_1+\cdots +a_kx_k\equiv b \pmod{n}$, with $\gcd(x_i,n)=t_i$ ($1\leq i\leq k$), where $a_1,t_1,\ldots,a_k,t_k, b,n$ ($n\geq 1$) are arbitrary integers. As a consequence, we derive necessary and sufficient conditions under which the above restricted linear congruence has no solutions. The number of solutions of this kind of congruence was first considered by Rademacher in 1925 and Brauer in 1926, in the special case of $a_i=t_i=1$ $(1\leq i \leq k)$. Since then, this problem has been studied, in several other special cases, in many papers; in particular, Jacobson and Williams [{\it Duke Math. J.} {\bf 39} (1972), 521--527] gave a nice explicit formula for the number of such solutions when $(a_1,\ldots,a_k)=t_i=1$ $(1\leq i \leq k)$. The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions.Comment: Journal of Number Theory, to appea

Classification of Minimal Separating Sets of Low Genus Surfaces

A minimal separating set in a connected topological space $X$ is a subset $L \subset X$ with the property that $X \setminus L$ is disconnected, but if $L^{\prime}$ is a proper subset of $L$, then $X \setminus L^{\prime}$ is connected. Such sets show up in a variety of contexts. For example, in a wide class of metric spaces, if we choose distinct points p and q, then the set of points x satisfying d(x, p) = d(x, q) is a minimal separating set. In this paper we classify which topological graphs can be realized as minimal separating sets in surfaces of low genus. In general the question of whether a graph can be embedded at all in a surface is a difficult one, so our work is partly computational. We classify graphs embeddings which are minimal separating in a given genus and write a computer program to find all such embeddings and their underlying graphs.Comment: 19 pages, 6 figure