Groups with ALOGTIME-hardWord Problems and PSPACE-complete CompressedWord Problems

We give lower bounds on the complexity of the word problem for a large class of non-solvable infinite groups that we call strongly efficiently non-solvable groups. This class includes free groups, Grigorchuk's group, and Thompson's groups. We prove that these groups have an NC1-hard word problem and that for some of them (including Grigorchuk's group and Thompson's groups) the compressed word problem (which is equivalent to the circuit evaluation problem) is PSPACE-complete.