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We prove that, in the l-ball of the Cayley graph of the braid group with n 3 strands, the proportion of rigid pseudo-Anosov braids is bounded below independently of l by a positive value. Proposition 2.1. The set of simple braids is in bijection with the set S n of permutations of n elements, via the canonical projection from B n to S n . Definition 2.2 (left-weighting). Let s 1 , s 2 be two simple braids in B n . We say that s 1 and s 2 are left-weighted, or that the pair (s 1 , s 2 ) isdoi:10.4171/ggd/406 fatcat:qnu5m6kc5vaxlflwiscktgrtaa