  We investigate the average-case complexity of decision problems for finitely
generated groups, in particular the word and membership problems. Using our
recent results on ``generic-case complexity'' we show that if a finitely
generated group $G$ has the word problem solvable in subexponential time and
has a subgroup of finite index which possesses a non-elementary word-hyperbolic
quotient group, then the average-case complexity of the word problem for $G$ is
linear time, uniformly with respect to the collection of all length-invariant
measures on $G$. For example, the result applies to all braid groups $B_n$.
