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A classification of a class of Buekenhout geometries exploiting amalgams and simple connectedness.
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4-dimensional football, fullerenes and diagram geometry
No full textAn n-fullerene is an n-dimensional cell complex where the stars of the points are (n - 1)-dimensional simplices and the 2-cells are pentagons or hexagons. We say that an n-fullerene is uniform if the number of hexagonal faces containing a given vertex p (and, when n > 3, contained in a given 3-face X on p) does not depend on the choice of p (and X). For instance, the dodecahedron, the truncated icosahedron (also called the football) and the tesselation of the euclidean plane in regular hexagons are uniform fullerenes. In this paper, we exploit notions and results of diagram geometry to classify finite uniform fullerenes. In particular, we prove that there is no four-dimensional analogue of the football. More precisely, we prove that there is just one simply connected 4-fullerene where the cells are truncated icosahedra, but it is obtained as a Grassmann geometry of a non-spherical (whence, infinite) Coxeter complex. Being infinite, that fullerene is not a polytope. (C) 2001 Elsevier Science B.V. All rights reserved
Shadow geometries and simple connectedness
No full textWe generalize results by Tits [22], Rees [13] and Rinauro [14], proving for large classes of 2-simply connected geometries with string diagrams that being 'thin at top' is equivalent to being obtainable as shadow geometries from 2-simply connected geometries with 'broom' diagrams (defined in Section 1.3 of this paper) or with completely disconnected diagrams or with 'complete' diagrams (defined in Section 1.3 of this paper). A non-simple-connectedness criterion is also obtained, as a by-product of the above. We apply that criterion to some geometries for the groups S4(3), S6 (2), O′N, G2(2), Aut(HJ), Aug(G2(4)) and Aut(Suz), proving that they cannot be simply connected. The inquiry begun in this paper is continued in [12], where automorphism groups of shadow geometries are investigated
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