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At the risk of being subjective (and possibly even argumentative), I feel like I should offer an answer to the implicit question, answered piquantly by Derek Holt: 'a large proportion of researchers in Geometric Group Theory appear to be interested only in the existence of algorithms and their complexity, rather than in actually using them to do ...


An algorithm for computing the hyperbolic "thinness" constant $\delta$ is described in the paper: Epstein, David B. A.; Holt, Derek F Computation in word-hyperbolic groups. Internat. J. Algebra Comput. 11 (2001), no. 4, 467–487. and I did implement it. It works OK on reasonably straightforward examples like surface groups, hyperbolic triangle groups, etc, ...

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