##
**The virtual cohomological dimension of the mapping class group of an orientable surface.**
*(English)*
Zbl 0592.57009

Let \(\Gamma =\Gamma^ s_{g,r}\) be the mapping class group of an orientable surface \(F=F^ s_{g,r}\) of genus \(g\) with \(s\) punctures and \(r\) boundary components. Thus each element of \(\Gamma\) is represented by an orientation preserving diffeomorphism of \(F\) which preserves the punctures of \(F\) individually and restricts to the identity on \(\partial F\). The group \(\Gamma\) is known to possess torsion-free subgroups of finite index, i.e. \(\Gamma\) is virtually torsion-free. The main result of the paper under review asserts that \(\Gamma\) has, in addition, cohomological properties parallelling those of the arithmetic groups.

Theorem 4.1. If \(2(g-1)+r+s\) is positive, the group \(\Gamma =\Gamma^ s_{g,r}\) is a virtual duality group (in the sense of Bieri-Eckmann) of dimension \(d=d(g,r,s)\) whose dualizing module \(H^ d(\Gamma, {\mathbb Z}\Gamma)\) is free abelian. The cohomological dimension d is \(4(g- 1)+2r+s\) if \(g>0\) and \(r+s>0\), it is 4(g-1)-1 if \(r=s=0\) and \(2r+s-3\) if \(g=0.\)

The groups for which \(2(g-1)+r+s\) is not positive are either trivial, infinite cyclic or isomorphic to \(\text{SL}(2,{\mathbb Z})\); in particular, they are virtual duality groups.

The main steps in the proof of the theorem are the following ones: First assume there is a boundary component. By plugging a boundary component with a punctured disc one obtains an embedding \(F^ s_{g,r}\hookrightarrow F^{s+1}_{g,r-1}\) which induces a projection of \(\Gamma^ s_{g,r}\) onto \(\Gamma^{s+1}_{g,r-1}\) with infinite cyclic kernel. This fact allows one to concentrate on the cases with \(r=0.\)

If \(r=0\) the group \(\Gamma^ s_ g=\Gamma^ s_{g,0}\) acts properly discontinuously on Teichmüller space \(T=T^ s_ g\) and the stabilizers of points are all finite. The space \(T\) is known to be homeomorphic to Euclidean space of dimension \(6(g-1)+2s\) whence each torsion-free subgroup of \(\Gamma^ s_ g\) has cohomological dimension bounded by \(6(g-1)+2s\). To obtain sharper results a contractible bordification \(\bar T\) of \(T\) with compact quotient \(\bar T/\Gamma\) is called for. A Borel-Serre bordification of \(T^ 0_ g\) has been constructed by W. Harvey using analytical methods [Ann. Math. Stud. 97, 241–251 (1981; Zbl 0461.30036)].

In this paper such a bordification \(W\) is built for the spaces \(T^ s_ g\) with \(s>0\) by a different, combinatorial method, and its boundary \(\partial W\) is shown to be homotopy equivalent to a wedge of spheres of dimension \(2(g-1)+s-1\) if \(g>0\), and s-4 if \(g=0\), \(s\geq 4\). (The group \(\Gamma^ s_ 0\) is trivial for \(s<4.)\) The properties of \(W\), in conjunction with known results from algebraic topology and cohomology theory of groups, imply the claim of Theorem 4.1 for the cases where \(s>0\). To establish the claim for \(g>1\), \(s=0\) the author makes use of the fact that the inclusion \(F^ 1_{g,0}\hookrightarrow F^ 0_{g,0}=F_ g\) induces a surjection of \(\Gamma^ 1_ g\) onto \(\Gamma^ 0_ g\) whose kernel is \(\pi_ 1(F_ g)\). Since \(\Gamma^ 0_ g\) is known to be virtually of finite cohomological dimension, this implies that \(\text{vcd}(\Gamma^ 0_ g)=\text{vcd}(\Gamma^ 1_ g)-2=4(g-1)-1\). To prove that \(\Gamma^ 0_ g\) is a duality group he takes recourse to Harvey’s construction of a bordification of \(T^ 0_ g\) mentioned above. (Alternatively, one could quote a theorem of R. Bieri [J. Pure Appl. Algebra 7, 35–51 (1976; Zbl 0322.20024)] asserting that a quotient of a duality group by a group of type FP is a duality group, provided the dimension of the quotient is finite.)

In addition to the bordification the author describes a spine \(Y\) of \(T\), a cell complex of dimension \(4(g-1)+s\) onto which \(T\) can be retracted \(\Gamma\)-equivariantly. \(Y\) has the lowest possible dimension in case \(g>0\) and \(s>0.\)

The construction of W and Y both make use of an ideal triangulation of \(T^ s_{g,r}\). This triangulation relies on a theorem of K. Strebel [J. Anal. Math. 19, 373–382 (1967; Zbl 0158.324)] according to which each point of \(T^ s_{g,r}\) corresponds uniquely to a quadratic differential on \(F^ s_{g,r}\) with special behaviour near punctures and boundary components.

Theorem 4.1. If \(2(g-1)+r+s\) is positive, the group \(\Gamma =\Gamma^ s_{g,r}\) is a virtual duality group (in the sense of Bieri-Eckmann) of dimension \(d=d(g,r,s)\) whose dualizing module \(H^ d(\Gamma, {\mathbb Z}\Gamma)\) is free abelian. The cohomological dimension d is \(4(g- 1)+2r+s\) if \(g>0\) and \(r+s>0\), it is 4(g-1)-1 if \(r=s=0\) and \(2r+s-3\) if \(g=0.\)

The groups for which \(2(g-1)+r+s\) is not positive are either trivial, infinite cyclic or isomorphic to \(\text{SL}(2,{\mathbb Z})\); in particular, they are virtual duality groups.

The main steps in the proof of the theorem are the following ones: First assume there is a boundary component. By plugging a boundary component with a punctured disc one obtains an embedding \(F^ s_{g,r}\hookrightarrow F^{s+1}_{g,r-1}\) which induces a projection of \(\Gamma^ s_{g,r}\) onto \(\Gamma^{s+1}_{g,r-1}\) with infinite cyclic kernel. This fact allows one to concentrate on the cases with \(r=0.\)

If \(r=0\) the group \(\Gamma^ s_ g=\Gamma^ s_{g,0}\) acts properly discontinuously on Teichmüller space \(T=T^ s_ g\) and the stabilizers of points are all finite. The space \(T\) is known to be homeomorphic to Euclidean space of dimension \(6(g-1)+2s\) whence each torsion-free subgroup of \(\Gamma^ s_ g\) has cohomological dimension bounded by \(6(g-1)+2s\). To obtain sharper results a contractible bordification \(\bar T\) of \(T\) with compact quotient \(\bar T/\Gamma\) is called for. A Borel-Serre bordification of \(T^ 0_ g\) has been constructed by W. Harvey using analytical methods [Ann. Math. Stud. 97, 241–251 (1981; Zbl 0461.30036)].

In this paper such a bordification \(W\) is built for the spaces \(T^ s_ g\) with \(s>0\) by a different, combinatorial method, and its boundary \(\partial W\) is shown to be homotopy equivalent to a wedge of spheres of dimension \(2(g-1)+s-1\) if \(g>0\), and s-4 if \(g=0\), \(s\geq 4\). (The group \(\Gamma^ s_ 0\) is trivial for \(s<4.)\) The properties of \(W\), in conjunction with known results from algebraic topology and cohomology theory of groups, imply the claim of Theorem 4.1 for the cases where \(s>0\). To establish the claim for \(g>1\), \(s=0\) the author makes use of the fact that the inclusion \(F^ 1_{g,0}\hookrightarrow F^ 0_{g,0}=F_ g\) induces a surjection of \(\Gamma^ 1_ g\) onto \(\Gamma^ 0_ g\) whose kernel is \(\pi_ 1(F_ g)\). Since \(\Gamma^ 0_ g\) is known to be virtually of finite cohomological dimension, this implies that \(\text{vcd}(\Gamma^ 0_ g)=\text{vcd}(\Gamma^ 1_ g)-2=4(g-1)-1\). To prove that \(\Gamma^ 0_ g\) is a duality group he takes recourse to Harvey’s construction of a bordification of \(T^ 0_ g\) mentioned above. (Alternatively, one could quote a theorem of R. Bieri [J. Pure Appl. Algebra 7, 35–51 (1976; Zbl 0322.20024)] asserting that a quotient of a duality group by a group of type FP is a duality group, provided the dimension of the quotient is finite.)

In addition to the bordification the author describes a spine \(Y\) of \(T\), a cell complex of dimension \(4(g-1)+s\) onto which \(T\) can be retracted \(\Gamma\)-equivariantly. \(Y\) has the lowest possible dimension in case \(g>0\) and \(s>0.\)

The construction of W and Y both make use of an ideal triangulation of \(T^ s_{g,r}\). This triangulation relies on a theorem of K. Strebel [J. Anal. Math. 19, 373–382 (1967; Zbl 0158.324)] according to which each point of \(T^ s_{g,r}\) corresponds uniquely to a quadratic differential on \(F^ s_{g,r}\) with special behaviour near punctures and boundary components.

Reviewer: R. Strebel

### MSC:

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |

20F38 | Other groups related to topology or analysis |

20J05 | Homological methods in group theory |

30F30 | Differentials on Riemann surfaces |

### Keywords:

mapping class group of an orientable surface; virtually torsion-free; arithmetic groups; virtual duality group; Teichmüller space; Borel- Serre bordification; quadratic differential### References:

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