A lattice in a residually non-Desarguesion \(\tilde A_2\)-building
(2017) Bulletin of the London mathematical society — Vol. 49, n° 2, p. 274-290 (2017)
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- Abstract
- Spherical and euclidean buildings have been classified by Tits and Weiss and are what is sometimes called “of algebraic origin” provided they are large enough. For spherical buildings a sufficient condition is that the rank is at least 3; for euclidean buildings, that the dimension is at least 3. Below these borders there are exotic buildings. More than just being exotic, two-dimensional euclidean buildings may even be residually exotic, meaning that the links of vertices (which are spherical buildings) are already exotic. Such examples have been long known by work of M.A. Ronan [in \textit{Buildings and the geometry of diagrams} (\textit{Como}, 1984), 242–248, Lecture Notes in Math., 1181, Springer, Berlin, 1986; MR0843395]. The present article affirmatively answers the natural question whether residually exotic buildings exist that in addition admit a uniform lattice, i.e., a cocompact group action with finite stabilizers. Specifically, the author describes an \(\tilde A_2\)-building with a vertex-regular group action whose vertex links are isomorphic to the Hughes plane of order 9. By work of D.I. Cartwright et al. [\textit{Geom. Dedicata} 47-2 (1993), 143–166; MR1232965], a vertex-regular lattice is parametrized by certain combinatorial data, called a triangle presentation compatible with a point-line correspondence. The proof of the main result therefore consists essentially of the tables in the appendix describing these data. However, the various steps of the computer search that led to it are interesting as well. The author describes and estimates a score that measures how close a point-line correspondence comes to admitting a compatible triangle presentation. He starts with a point-line correspondence that is a correlation. It is known that correlations cannot admit a triangle presentation but they have a relatively high score. A greedy computer search then tries to improve the score until hitting a local maximum. Ironically this strategy seems to not quite work, but a change to the code that inadvertently weakens the greediness of the algorithm leads to success. \textit{Stefan Witzel}
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Radu, N. (2017). A lattice in a residually non-Desarguesion \(\tilde A_2\)-building. Bulletin of the London mathematical society, 49(2), 274-290. https://hdl.handle.net/2078.5/58682 (Original work published 2017)