Algebraic mapping-class groups of orientable surfaces with boundaries

Abstract

Let Sg,b,p denote a surface which is connected, orientable, has genus g, has b boundary components, and has p punctures. Let Σg,b,p denote the fundamental group of Sg,b,p. We define the algebraic mapping-class group of Sg,b,p, denoted by Outg,b,p, and observe that topologists have shown that Outg,b,p is naturally isomorphic to the topological mapping-class group of Sg,b,p. We study the algebraic version (formula presented) of Mess’s exact sequence that arises from filling in the interior of the (b + 1)st boundary component of Sg,b+1,p. Here Outg,b⊥1,p is the subgroup of index b + 1 in Outg,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{\smash{\scriptscriptstyle\smile}}}{\Sigma } _{g,b,p} is trivial. If (g, b, p) is (0, 0, 2) or (1, 0, 0), then \overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\smile}}}{\Sigma } _{g,b,p} is infinite cyclic. In all other cases, \overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\smile}}}{\Sigma } _{g,b,p} is the fundamental group of the unit-tangent bundle of a suitably metrized Sg,b,p, and, hence, \overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\smile}}}{\Sigma } _{g,b,p} is an extension of an infinite cyclic, central subgroup by Σg,b,p. We give a description of the conjugation action of Outg,b⊥1,p on \overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\smile}}}{\Sigma } _{g,b,p} in terms of the following three ingredients: an easily-defined action of Outg,b⊥1,p on Σg,b+1,p; the natural homomorphism Σg,b+1,p → \overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\smile}}}{\Sigma } _{g,b,p}; and, a twisting-number map Σg,b+1,p → ℤ that we define. The work of many authors has produced aesthetic presentations of the orientation-preserving mapping-class groups Outg,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{\smash{\scriptscriptstyle\smile}}}{\Sigma } _{g,0,0} of the map 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, Outg,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{\smash{\scriptscriptstyle\smile}}}{\Sigma } _{g,0,0} is the kernel of the natural map between these groups. What we study are presentations for \overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\smile}}}{\Sigma } _{g,0,0}, both as a group and as an Outg,1,0-group, and related topics.