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Then U1 ∪U2 is contained in u1 u2 , so cw (u)−cw (u1 )−cw (u2 ) ≥ 0. Conversely, let U be a maximal disjoint configuration of copies of w in u1 u2 . Then either U contains one copy of w which intersects the juncture, or else it is disjoint from the juncture and decomposes as U = U1 ∪U2 . Hence cw (u)−cw (u1 )−cw (u2 ) ≤ 1 if s = 1 and cw (u) − cw (u1 ) − cw (u2 ) ≤ 0 otherwise. It follows that D(Hw ) ≤ 3(|w| − 1). One cannot do better than O(|w|) in general, as an example like w = abababababa shows.

E. it is a homomorphism. Otherwise, if w = w1 w2 is a reduced expression, hw (w) = 1 whereas hw (w1 ) = hw (w2 ) = 0. This proves the first statement. Let u, v ∈ F be reduced. Then we can uniquely write u = u′ x, v = x−1 v ′ where ′ ′ u v is the reduced representative of uv. Let s1 , s2 , s3 be the signs of the reduced expressions u′ x, x−1 v ′ , u′ v ′ respectively. We calculate hw (uv) − hw (u) − hw (v) = hw (uv) − hw (u) − hw (v) − hw (u′ ) + hw (u′ ) − hw (v ′ ) + hw (v ′ ) + hw (x) − hw (x−1 ) = (0 or s3 ) − (0 or s1 ) − (0 or s2 ) After possibly replacing w with w−1 and are only nine possibilities for (s1 , s2 , s3 ): 0 for 1 for |hw (uv) − hw (u) − hw (v)| ≤ 2 for 3 for reversing the order of the strings, there (0, 0, 0) (1, 0, 0), (0, 0, 1), (1, −1, 0), (1, 0, 1) (1, 1, 0), (1, 1, 1), (1, 0, −1) (1, 1, −1) Case ((1, 0, −1)).

In other words, SP admits a hyperbolic metric. Moreover, the only disk pieces are components of Si mapping with degree a multiple of ni to γi , in the case gi is torsion. Since fP is proper, it has a well-defined degree. Since fP−1 (γP ) is equal to ∂S, the degree is n.