The author proves that every split extension of a finitely generated abelian or word-hyperbolic group [in the sense of \textit{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 \textit{J. Cannon}, \textit{D. Epstein}, \textit{D. Holt}, \textit{S. Levy}, \textit{M. Paterson} and \textit{W. Thurston}, Word processing in groups (1992; Zbl 0764.20017), and a preprint of \textit{S. M. Gersten}, Isodiametric and isoperimetric inequalities in group extensions (1991)].