  We consider three important and well-studied algorithmic problems in group
theory: the word, geodesic, and conjugacy problem. We show transfer results
from individual groups to graph products. We concentrate on logspace complexity
because the challenge is actually in small complexity classes, only. The most
difficult transfer result is for the conjugacy problem. We have a general
result for graph products, but even in the special case of a graph group the
result is new. Graph groups are closely linked to the theory of Mazurkiewicz
traces which form an algebraic model for concurrent processes. Our proofs are
combinatorial and based on well-known concepts in trace theory. We also use
rewriting techniques over traces. For the group-theoretical part we apply
Bass-Serre theory. But as we need explicit formulae and as we design concrete
algorithms all our group-theoretical calculations are completely explicit and
accessible to non-specialists.
