Surface quotients of hyperbolic buildings

Details Surface quotients of hyperbolic buildings Surface quotients of hyperbolic buildings

2010-07-29

arxiv.org/abs/1007.5140v2

Int Math Res Notices 2012 (Issue 2): 437-477

Mathematics

Group Theory

23 pages, 4 figures. Version 2 incorporates referee's suggestions including new Section 7 discussing relationships between our

Scientific paper

10.1093/imrn/rnr028

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 {\Gamma} in Aut(I(p,v)) such that {\Gamma}\I(p,v) is a compact orientable surface of genus g. Surprisingly, the existence of {\Gamma} depends upon the value of v. The remaining cases lead to open questions in tessellations of surfaces and in number theory. Our construction of {\Gamma}, together with a theorem of Haglund, implies that for p>=6, every uniform lattice in Aut(I) contains a surface subgroup. We use elementary group theory, combinatorics, algebraic topology, and number theory.

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