We identify the images of the comparision maps from ordinary homology and Sobolev homology, respectively, to the `1-homology of a wordhyperbolic group with coefficients in complete normed modules. The underlying idea is that there is a subdivision procedure for singular chains in negatively curved spaces that is much more efficient (in terms of the `1-norm) than barycentric subdivision. The res...

We implement various computations in the braid groups via practically efficient and theoretically optimized algorithms whose pseudo-codes are provided. The performance of an actual implementation under various choices of parameters is listed.

Using HNN extensions of the Boone-Britton group, a group E is obtained which simulates Turing machine computation in linear space and cubic time. Space in E is measured by the length of words, and time by the number of substitutions of defining relators and conjugations by generators required to convert one word to another. The space bound is used to derive a PSPACE-complete problem for a topol...

Let G be a finite soluble group given by a reduced confluent polycyclic presentation. Represent group elements by reduced words. Then there exists an algorithm for multiplying two group elements which has subexponential running time and requires polynomial space. Moreover, given the prime factorisation of the group order, the problem of multiplying two group elements is probabilistically polyno...

We study the compressed word problem: a variant of the word problem for finitely generated groups where the input word is given by a context-free grammar that generates exactly one string. We show that finite extensions and free products preserve the complexity of the compressed word problem. Also, the compressed word problem for a graph group can be solved in polynomial time. These results all...