The word problem for geometrically finite groups
Let \(<X:R>\) be a presentation of a group G, with \(G=F/N\), where F is a free group with basis X and N is the normal closure of R in F. For \(n\in {\mathbb{N}}\) let R(n) be the set of all words in X which are conjugate to some \(r\in R\) by some reduced word of length at most n. Let F(n) be the set of initial segments of words in R(n). On F(n) there is an equivalence relation \(\sim_ n\) where \(w_ 1w_ 2\sim_ nw_ 1w_ 0w_ 2\) whenever both are in F(n) and \(w_ 0\) is in R(n). Let f be any function \(f: {\mathbb{N}}\to {\mathbb{N}}\). The presentation \(<X:R>\) is said to be f-decidable if given any element in F represented by a word w of length m then \(w\in N\) iff \(w\sim_ n1\) for \(n=f(m)\). The Todd- Coxeter algorithm shows that any presentation is f-decidable for some f. But in general f needs not to be recursive. The word problem for G is decidable iff G possesses a presentation such that this presentation is f-decidable for some recursive f. In the paper under review the authors prove the following: (i) Let G be a cocompact group of isometries of hyperbolic or euclidean space of dimension \(d\geq 2\) and \(<X:R>\) be the presentation obtained from a Dirichlet region for G, then there is a natural number s such that \(<X:R>\) is f-decidable for \(f(m)=sm^ 2\). (ii) Let G be a discrete group of isometries of hyperbolic d-space such that either \(d\geq 2\) and H/G has finite volume or \(d=3\) and G is geometrically finite. Let \(<X:R>\) be a presentation as before, then \(<X:R>\) is f-decidable for \(f(m)=[e^{sm}]\) and suitable s.
scientific article; zbMATH DE number 3957424
scientific article