TY - JOUR
T1 - Three-dimensional maps and subgroup growth
AU - Bottinelli, Rémi
AU - Ciobanu, Laura
AU - Kolpakov, Alexander
N1 - Funding Information:
The authors gratefully acknowledge the support received from the University of Neuchâtel Overhead Grant No. 12.8/U.01851. R.B. and A.K. were also supported by FN PP00P2-144681/1, and L.C. was supported by FN PP00P2-170560 projects of the Swiss National Science Foundation. The authors greatly appreciate the fruitful discussions with the On-line Encyclopaedia of Integer Sequences—OEIS community, who helped identify integer sequences in this manuscript and improve its exposition. The computations were performed using the “Cervino” computational cluster of the Computer Science Department at the University of Neuchâtel.
Publisher Copyright:
© 2021, The Author(s).
PY - 2022/7
Y1 - 2022/7
N2 - 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 Δ += Z2∗ Z2∗ Z2 via a simple bijection between pavings and finite index subgroups which can be deduced from the action of Δ + 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 Δ +. Computational experiments performed with software designed by the authors provide some statistics about the topology and combinatorics of pavings on n≤ 16 darts.
AB - 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 Δ += Z2∗ Z2∗ Z2 via a simple bijection between pavings and finite index subgroups which can be deduced from the action of Δ + 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 Δ +. Computational experiments performed with software designed by the authors provide some statistics about the topology and combinatorics of pavings on n≤ 16 darts.
UR - http://www.scopus.com/inward/record.url?scp=85108807120&partnerID=8YFLogxK
U2 - 10.1007/s00229-021-01321-7
DO - 10.1007/s00229-021-01321-7
M3 - Article
C2 - 35726247
AN - SCOPUS:85108807120
SN - 0025-2611
VL - 168
SP - 549
EP - 570
JO - Manuscripta Mathematica
JF - Manuscripta Mathematica
ER -