# Surface quotients of hyperbolic buildings

*Algebra/Geometry/Topology Seminar*

*by Anne Thomas*

*Institution:*University of Sydney

*Date: Mon 15th November 2010*

*Time: 2:00 PM*

*Location: Room 213, Richard Berry Building, The University of Melbourne*

*Abstract*: Let I(p,v) be Bourdon's building, the unique simply-connected 2-complex such that all 2-cells are regular right-angled hyperbolic p-gons and the link at each vertex is the complete bipartite graph K(v,v). We investigate and mostly determine the set of triples (p,v,g) for which there exists a uniform lattice in Aut(I(p,v)) such that the quotient of I(p,v) by this lattice is a compact orientable surface of genus g. Surprisingly, the existence of such a lattice depends upon the value of v. The remaining cases lead to open questions in tessellations of surfaces and in number theory. Our construction, together with a theorem of Haglund, implies that for p>=6, every uniform lattice in Aut(I(p,v)) contains a surface subgroup. We use elementary group theory, combinatorics, algebraic topology, and number theory. This is joint work with David Futer.

*For More Information:* contact: Craig Westerland. email: c.westerland@ms.unimelb.edu.au