Surface quotients of hyperbolic buildings
by Anne Thomas
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.
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