  Berenstein and Kirillov have studied the action of Bender-Knuth moves on
semistandard tableaux. Losev has studied a cactus group action in
Kazhdan-Lusztig theory; in type $A$ this action can also be identified in the
work of Henriques and Kamnitzer. We establish the relationship between the two
actions. We show that the Berenstein-Kirillov group is a quotient of the cactus
group. We use this to derive previously unknown relations in the
Berenstein-Kirillov group. We also determine precise implications between
subsets of relations in the two groups, which yields a presentation for cactus
groups in terms of Bender-Knuth generators.
