The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relatively recently. In this paper, we address the average-case time complexity of the word problem in several classes of groups and show that it is often the case that the average-case complexity is linear with respect to the length of an input word. The classes of groups that we consider include groups of matrices over rationals (in particular, polycyclic groups), some classes of solvable groups, as well as free products. Along the way, we improve several bounds for the worst-case complexity of the word problem in groups of matrices, in particular in nilpotent groups. For free products, we also address the average-case complexity of the subgroup membership problem and show that it is often linear, too. Finally, we discuss complexity of the identity problem that has not been considered before.

翻译：群论算法的最坏情况复杂度已研究良久。通用情况复杂度（即随机输入下的复杂度）是较近期引入并研究的。本文探讨了几类群中词问题的平均情况时间复杂度，并证明在许多情况下，其平均复杂度与输入词长度呈线性关系。我们研究的群类包括有理数矩阵群（特别是多循环群）、某些可解群类以及自由积。在此过程中，我们改进了矩阵群（尤其是幂零群）中词问题最坏情况复杂度的若干界。对于自由积，我们还探讨了子群成员问题的平均复杂度，并证明其同样常呈线性。最后，我们讨论了此前未被考虑过的单位元问题的复杂度。