# zbMATH — the first resource for mathematics

Growth functions of groups of surfaces. (English. Russian original) Zbl 0861.20034
Math. Notes 58, No. 5, 1156-1165 (1995); translation from Mat. Zametki 58, No. 5, 681-693 (1995).
The main result is a formula for the growth function (here the generating function $$\sum d_nz^n$$, where $$d_n$$ is the number of the elements in a group $$G$$, whose minimal presentation as words in the alphabet $$X=\{a_1,\dots,a_n,a^{-1}_1,\dots,a^{-1}_n\}$$ has length $$n$$) of the fundamental group of a closed orientable surface of genus $$g$$: $$G=\langle X\mid\prod^g_{i=1}[a_i,b_i]=1\rangle$$. The proof uses the term rewriting approach and standard formula for calculating generating series in a confluent term rewriting system.

##### MSC:
 20F05 Generators, relations, and presentations of groups 57M05 Fundamental group, presentations, free differential calculus 20F34 Fundamental groups and their automorphisms (group-theoretic aspects) 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 68R15 Combinatorics on words 68Q42 Grammars and rewriting systems
Full Text:
##### References:
 [1] R. I. Grigorchuk, ”On growth in group theory,”Proceedings of the ICM Kyoto 90,1, 325–338 (1990). · Zbl 0749.20016 [2] J. W. Cannon,The Growth of the Closed Groups of Surfaces and the Compact Hyperbolic Groups, Preprint, Boston (1979). [3] R. I. Grigorchuk, ”Growth functions, rewriting systems, and the Euler characteristic,”Mat. Zametki [Math. Notes],58, No. 5, 653–668 (1995). · Zbl 0860.68084 [4] M. Stoll,Some Group Presentations with Rational Growth, Preprint, Boston (1994). · Zbl 1112.11307 [5] D. E. Cohen, ”String rewriting: a survey for group theorists,” in:Geometric Group Theory, Vol. 1, Sussex (1991);London Math. Soc. Lect. Note Series,181, 37–47 (1993). · Zbl 0833.20037 [6] D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson, and W. P. Thurston,Word Processing in Groups, Jones and Bartlett, Boston (1992). · Zbl 0764.20017 [7] N. Bourbaki,Groupes et algĂ¨bres de Lie, Hermann, Paris (1968). · Zbl 0186.33001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.