  We consider oriented knots and links in a handlebody of genus $g$ through
appropriate braid representatives in $S^3$, which are elements of the braid
groups $B_{g,n}$. We prove a geometric version of the Markov theorem for braid
equivalence in the handlebody, which is based on the $L$-moves. Using this we
then prove two algebraic versions of the Markov theorem. The first one uses the
$L$-moves. The second one uses the Markov moves and conjugation in the groups
$B_{g,n}$. We show that not all conjugations correspond to isotopies.
