1,039 research outputs found

    On semi-finite hexagons of order (2,t)(2, t) containing a subhexagon

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    The research in this paper was motivated by one of the most important open problems in the theory of generalized polygons, namely the existence problem for semi-finite thick generalized polygons. We show here that no semi-finite generalized hexagon of order (2,t)(2,t) can have a subhexagon HH of order 22. Such a subhexagon is necessarily isomorphic to the split Cayley generalized hexagon H(2)H(2) or its point-line dual HD(2)H^D(2). In fact, the employed techniques allow us to prove a stronger result. We show that every near hexagon S\mathcal{S} of order (2,t)(2,t) which contains a generalized hexagon HH of order 22 as an isometrically embedded subgeometry must be finite. Moreover, if H≅HD(2)H \cong H^D(2) then S\mathcal{S} must also be a generalized hexagon, and consequently isomorphic to either HD(2)H^D(2) or the dual twisted triality hexagon T(2,8)T(2,8).Comment: 21 pages; new corrected proofs of Lemmas 4.6 and 4.7; earlier proofs worked for generalized hexagons but not near hexagon

    Surface quotients of hyperbolic buildings

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    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.Comment: 23 pages, 4 figures. Version 2 incorporates referee's suggestions including new Section 7 discussing relationships between our constructions, previous examples, and surface subgroups. To appear in Int. Math. Res. No
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