ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

6. Topology

1991 MS Classification: 20, 51, 57 Rogmann, Sascha, Heinrich-Heine-Universität Düsseldorf, Germany Growth functions of tessellations. Let X be a topological space tiled by a family T = (T_{λ})_{λ∈Λ} of tiles, μ ∈ Λ, S_{0} := {T_{μ}}. Define inductively shells around T_{μ} by S_{m+1} := {T_{λ} | T_{λ} ∉ S_{0} ∪ … ∪ S_{m} ∧ T_{λ} ∩ (⋃_{P ∈S_{m}} P) ≠ ∅} for all m ≥ 0. If all S_{m} are finite we can define the growth series of (X, T ) around T_{μ} by f_{(X,T ),T_{μ}}(x) := ∑_{m=0}^{∞} #S_{m}x^{m} ∈ ℤ[[x]]. Tessellations of hyperbolic space coming from discrete subgroups of Iso(ℍ^{2}) and Iso(ℍ^{3}) have been looked at concerning their growth series. The growth series I have found are rational and I examined their symmetry and connection with the Euler characteristic. The hyperbolic tessellations have been displayed by means of color plots and anaglyphs. Reference: W. Floyd, and S. Plotnick, Growth functions on Fuchsian groups and the Euler characteristic, Invent. math. 88 (1987), 1–29.