Simplicial representations of surface mapping class groups (joint with Athanase Papadopoulos)

In recent joint work with Athanase Papadopoulos, we have begun the study of a flag complex which is naturally associated to the Thurston theory of surface diffeomorphisms for compact connected orientable surfaces with boundary ([5]). The various pieces of the Thurston decomposition of a surface diffeomorphism, thick domains and annular or thin domains, fit into this flag complex, which we call the complex of domains. One of our main results about the complex of domains is the computation of the group of automorphisms of this complex. Unlike the celebrated complex of curves introduced by Harvey ([1]), for which, for all but a finite number of exceptional surfaces, by the works of Ivanov ([2]), Korkmaz ([3]), and Luo ([4]), all automorphisms are geometric (i.e. induced by homeomorphisms), the complex of domains has nongeometric automorphisms, provided the surface in question has at least two boundary components. These nongeometric automorphisms of the complex of domains are associated to natural "biperipheral edges" of the complex which are associated to biperipheral pairs of pants on the surface in question. We construct a natural quotient complex of the complex of domains, which we call the truncated complex of domains, by collapsing biperipheral edges to vertices and, thereby, reduce the computation in question to computing the group of automorphisms of the truncated complex of domains. Finally, we prove that the group of automorphisms of the truncated complex of domains is the extended mapping class group of the surface in question and, obtain, thereby, a complete description of the group of automorphisms of the complex of domains.

[1] W. J. Harvey, Geometric structure of surface mapping class groups, Homological group theory (Proc. Sympos., Durham, 1977), pp. 255-269, London Math. Soc. Lecture Note Ser., 36, Cambridge Univ. Press, Cambridge-New York, 1979

[2] N. V. Ivanov, Automorphisms of Teichmüller modular groups, Lecture Notes in Math., No. 1346, Springer-Verlag, Berlin and New York, 1988, 199-270.

[3] M. Korkmaz, Automorphisms of complexes of curves on punctured spheres and on punctured tori, Topology and its Applications 95 no. 2 (1999), 85-111.

[4] F. Luo, Automorphisms of the complex of curves, Topology 39 (2000), no. 2, 283-298.

[5] J. D. McCarthy and A. Papadopoulos, Tilings of surfaces: the complex of domains, monograph in preparation.