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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.

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: DOI
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