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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
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##### References:
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