We define the algebraic mapping-class group of S_g,b,p, denoted by Out_g,b,p, and observe that topologists have shown that Out_g,b,p is naturally isomorphic to the topological mapping-class group of S_g,b,p.

We study the algebraic version

$ 1 \to \overset{\lower0.5em\hbox{$ 1 \to \overset{\lower0.5em\hbox{

of Mess’s exact sequence that arises from filling in the interior of the (b + 1)st boundary component of S_g,b+1,p.

Here Out_g,b⊥1,p is the subgroup of index b + 1 in Out_g,b+1,p that fixes the (b + 1)st boundary component.

If (g, b, p) is (0, 0, 0) or (0, 0, 1), then $ \overset{\lower0.5em\hbox{$ \overset{\lower0.5em\hbox{ is trivial. If (g, b, p) is (0, 0, 2) or (1, 0, 0), then $ \overset{\lower0.5em\hbox{$ \overset{\lower0.5em\hbox{ is infinite cyclic. In all other cases, $ \overset{\lower0.5em\hbox{$ \overset{\lower0.5em\hbox{ is the fundamental group of the unit-tangent bundle of a suitably metrized S_g,b,p, and, hence, $ \overset{\lower0.5em\hbox{$ \overset{\lower0.5em\hbox{ is an extension of an infinite cyclic, central subgroup by Σ_g,b,p.

We give a description of the conjugation action of Out_g,b⊥1,p on $ \overset{\lower0.5em\hbox{$ \overset{\lower0.5em\hbox{ in terms of the following three ingredients: an easily-defined action of Out_g,b⊥1,p on Σ_g,b+1,p; the natural homomorphism Σ_g,b+1,p → $ \overset{\lower0.5em\hbox{$ \overset{\lower0.5em\hbox{ ; and, a twisting-number map Σ_g,b+1,p → ℤ that we define.

The work of many authors has produced aesthetic presentations of the orientation-preserving mapping-class groups Out_g,b,p⁺ with b+p ≤ 1, using the DLH generators. Within the program of giving algebraic proofs to algebraic results, we apply our machinery to give an algebraic proof of a relatively small part of this work, namely that the kernel $ \overset{\lower0.5em\hbox{$ \overset{\lower0.5em\hbox{ of the map

_boxclose_boxclose_boxclose_boxclose_boxclose_boxclose_boxclose^ + Out_g,0,0

#xA; Out_{g,1,0}^ + \to Out_{g,0,0}

is the normal closure in Out_g,1,0⁺ of Matsumoto’s A-D word (in the DLH generators).

From the algebraic viewpoint, Out_g,1,0 is the group of those automorphisms of a rank-2g free group which fix or invert a given genus g surface relator, Outg,0,0 is the group of outer automorphisms of the genus g surface group, and $ \overset{\lower0.5em\hbox{$ \overset{\lower0.5em\hbox{ is the kernel of the natural map between these groups. What we study are presentations for $ \overset{\lower0.5em\hbox{$ \overset{\lower0.5em\hbox{ , both as a group and as an Out_g,1,0-group, and related topics.