Bridge numbers of knots in the page of an open book

Given any closed, connected, orientable $3$–manifold and integers $g\geq g(M), D > 0$,weshowtheexistenceofknotsin$M$whosegenus$g$ bridge number is greater than $D$. These knots lie in a page of an open book decomposition of $M$, and the proof proceeds by examining the action of the map induced by the monodromy on the arc and curve complex of a page. A corollary is that there are Berge knots of arbitrarily large genus one bridge number.