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Combings of semidirect products and 3-manifold groups. (English) Zbl 0783.20020

The author proves that every split extension of a finitely generated abelian or word-hyperbolic group [in the sense of M. Gromov, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)] by an asynchronously combable group is asynchronously combable as well as every split extension of a word-hyperbolic group by an asynchronously automatic group is asynchronously automatic.
Constructing a normal form of an efficient geometric nature for elements in \(\pi_ 1(M)\) of any compact 3-manifold \(M\) that satisfies Thurston’s geometrization conjecture, the author proves that this \(\pi_ 1(M)\) is asynchronously combable, i.e. its normal form satisfies the asynchronous fellow-traveller property. As a corollary, it follows that such \(\pi_ 1(M)\) satisfies an exponential isoperimetric inequality and a linear isodiametric inequality [see also J. Cannon, D. Epstein, D. Holt, S. Levy, M. Paterson and W. Thurston, Word processing in groups (1992; Zbl 0764.20017), and a preprint of S. M. Gersten, Isodiametric and isoperimetric inequalities in group extensions (1991)].

MSC:

20F65 Geometric group theory
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
57M50 General geometric structures on low-dimensional manifolds
57M05 Fundamental group, presentations, free differential calculus
20F05 Generators, relations, and presentations of groups
20E22 Extensions, wreath products, and other compositions of groups
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References:

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