Abstract
We survey results about computational complexity of the word problem in groups, Dehn functions of groups and related problems.
Highlights
The word and conjugacy problems are the most classical algorithmic problem for groups going back to the work of Dehn and Tietze at the beginning of the twentieth century
We present an algebraic characterization of groups with word problem in NP from [22] and groups with word problem in PSPACE from [183], discuss examples of groups with NP-complete [210] and coNP-complete [21] word problems
In [192], we showed that the group a, b, t1, t2 | ti ati−1 = ab, ti b = bti, i = 1, 2 has cubic Dehn function, linear isodiametric function and non- connected asymptotic cones
Summary
The word and conjugacy problems are the most classical algorithmic problem for groups going back to the work of Dehn and Tietze at the beginning of the twentieth century. There are very many papers devoted to these problems for various classes of groups Different aspects of these problems are discussed in many books and surveys (see, for example, [24,35,50,72,78,91,137,147,162,181,187,207,208]). Several new methods are used in the proofs of these results and our goal in this survey is to give as gentle as possible an introduction to these results and methods To this end, we often present not the results in their full generality but their easier to explain approximations. There is a nice recent survey by Shpilrain [219] where at least some of these topics are discussed
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