1,039 research outputs found
On semi-finite hexagons of order containing a subhexagon
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 can have a subhexagon of order .
Such a subhexagon is necessarily isomorphic to the split Cayley generalized
hexagon or its point-line dual . In fact, the employed
techniques allow us to prove a stronger result. We show that every near hexagon
of order which contains a generalized hexagon of
order as an isometrically embedded subgeometry must be finite. Moreover, if
then must also be a generalized hexagon, and
consequently isomorphic to either or the dual twisted triality hexagon
.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
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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