Sur les immeubles triangulaires et leurs automorphismes. (On triangle buildings and their automorphism.) (French) Zbl 1142.51011
J. Tits [Finite geometries, buildings, and related topics, Pap. Conf., Pingree Park/CO (USA) 1988, 17–28 (1990; Zbl 0744.51013)] showed that the vertices of triangle buildings (that is, affine buildings of type \(\widetilde{A}_2\)) are naturally coloured by their balls of radius 2 in case the building has order 2 and the first author [Ann. Fac. Sci. Toulouse, VI. Sér., Math. 9, No. 4, 575–603 (2000; Zbl 1003.51007)] demonstrated that such a colouring can be freely chosen.
In the paper under review the authors extend these results to order \(\geq 3\). They consider for a family \(P\) of projective planes of order \(q\geq 2\) the space \(E_P\) of all isomorphism classes of triangle buildings all whose links are associated with projective planes in \(P\). A triangle building itself is considered an element of \(E_P\) and a ball in \(E_P\) is a ball in one of its constituent triangle buildings. With this notation the authors obtain the following results.
(Colouring lemma) If \(B^2\) is a ball in \(E_P\) of radius 2 (which is classic in case \(q=3\) or \(q=4\)) and \(B_0\) is a ball in \(E_P\) of radius \(R\geq 1\) and centre \(O\), then there exists a ball \(B_1\) in \(E_P\) of radius \(R+1\) and the same centre \(O\) such that no ball of radius 2 and centre at a vertex on the sphere of radius \(R-1\) and centre \(O\) is isomorphic to \(B^2\).
(Surgery lemma) Let \(B\) be a ball in \(E_P\). Then for each ball \(B'\) in \(E_P\) different from \(B\) there exists a building \(\Delta\) in \(E_P\) that contains both \(B\) and \(B'\). Furthermore, \(B'\) can be positioned in a certain way relative to \(B\) in \(\Delta\). The building \(\Delta\) is constructed from a building containing \(B\) by pasting in a certain cone containing \(B'\).
As a consequence it follows that for each projective plane \(P\) the set \(E_P\) has the cardinality of the reals.
The authors further consider the set \(\Lambda_P\) of all isomorphism classes of triangle buildings in \(E_P\) with a basepoint. They show that \(\Lambda_P\) is equipped with a locally compact Hausdorff topology and, defining a natural cellular lamination on \(\Lambda_P\), that \(\Lambda_P\) is topologically transitive. Thus there exists a triangle building that contains all isomorphism types of balls of finite radius. Distinguishing the cases \(q=2\) and \(q\geq 3\) the authors show in the last two sections that a generic building in \(E_P\) has trivial automorphism group.
In the paper under review the authors extend these results to order \(\geq 3\). They consider for a family \(P\) of projective planes of order \(q\geq 2\) the space \(E_P\) of all isomorphism classes of triangle buildings all whose links are associated with projective planes in \(P\). A triangle building itself is considered an element of \(E_P\) and a ball in \(E_P\) is a ball in one of its constituent triangle buildings. With this notation the authors obtain the following results.
(Colouring lemma) If \(B^2\) is a ball in \(E_P\) of radius 2 (which is classic in case \(q=3\) or \(q=4\)) and \(B_0\) is a ball in \(E_P\) of radius \(R\geq 1\) and centre \(O\), then there exists a ball \(B_1\) in \(E_P\) of radius \(R+1\) and the same centre \(O\) such that no ball of radius 2 and centre at a vertex on the sphere of radius \(R-1\) and centre \(O\) is isomorphic to \(B^2\).
(Surgery lemma) Let \(B\) be a ball in \(E_P\). Then for each ball \(B'\) in \(E_P\) different from \(B\) there exists a building \(\Delta\) in \(E_P\) that contains both \(B\) and \(B'\). Furthermore, \(B'\) can be positioned in a certain way relative to \(B\) in \(\Delta\). The building \(\Delta\) is constructed from a building containing \(B\) by pasting in a certain cone containing \(B'\).
As a consequence it follows that for each projective plane \(P\) the set \(E_P\) has the cardinality of the reals.
The authors further consider the set \(\Lambda_P\) of all isomorphism classes of triangle buildings in \(E_P\) with a basepoint. They show that \(\Lambda_P\) is equipped with a locally compact Hausdorff topology and, defining a natural cellular lamination on \(\Lambda_P\), that \(\Lambda_P\) is topologically transitive. Thus there exists a triangle building that contains all isomorphism types of balls of finite radius. Distinguishing the cases \(q=2\) and \(q\geq 3\) the authors show in the last two sections that a generic building in \(E_P\) has trivial automorphism group.
Reviewer: Günter F. Steinke (Christchurch)
MSC:
| 51E24 | Buildings and the geometry of diagrams |
| 51E20 | Combinatorial structures in finite projective spaces |
| 81R60 | Noncommutative geometry in quantum theory |
References:
| [1] | Artin, E.: Algèbre Géométrique. Paris (1978) |
| [2] | Barré, S.: Polyèdres finis de dimension 2 à courbure négative ou nulle et de rang 2. Ann. de l’Inst. Fourier T45 (1995) |
| [3] | Barré, S.: Polyèdres de rang deux. Thèse, ENS Lyon (1996) |
| [4] | Barré, S.: Immeubles de Tits triangulaires exotiques. Ann. Fac. Sci. Toulouse Math. (6) 9(4), 575–603 (2000) · Zbl 1003.51007 |
| [5] | Barré, S., Pichot, M.: Existence d’immeubles triangulaires quasi-périodiques (prépublication) · Zbl 1219.51010 |
| [6] | Bridson, M., Haefliger, A.: Metric Spaces of Non-positive Curvature, vol. 319. Grundlehren der Mathematischen Wissenschaften, Berlin (1999) · Zbl 0988.53001 |
| [7] | Brown, K.S.: Buildings. Springer Verlag (1989) |
| [8] | Connes, A.: Sur la théorie non commutative de l’intégration. Algèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978), Lecture Notes in Math., vol. 725, pp. 19–143, Springer, Berlin (1979) |
| [9] | Haglund F. (2002). Existence, unicité et homogénéité de certains immeubles hyperboliques. Math. Z. 242(1): 97–148 · Zbl 1052.51009 · doi:10.1007/s002090100309 |
| [10] | Haglund F. and Paulin F. (2003). Construction arborescente d’immeubles, Math. Ann. 325: 137–164 · Zbl 1025.51014 |
| [11] | Moorhouse, G.E.: On projective planes of order less than 32. (English summary) Finite geometries, groups, and computation, pp. 149–162. Walter de Gruyter, Berlin (2006) · Zbl 1100.51004 |
| [12] | Pansu P. (1998). Formules de Matsushima, de Garland et propriété (T) pour des groupes agissant sur des espaces symétriques ou des immeubles. Bull. Soc. math. France 126: 107–139 · Zbl 0933.22009 |
| [13] | Pichot M. (2007). Espaces mesurés singuliers fortement ergodiques et structures métriques-mesurées. Ann. Inst. Fourier 57(1): 1–43 |
| [14] | Pichot M. (2007). Sur la théorie spectrale des relations d’équivalence mesurées. J. Inst. Math. Jussieu 6(3): 453–500 · Zbl 1135.37007 · doi:10.1017/S1474748007000096 |
| [15] | Ronan, M.A.: A construction of buildings with no rank 3 residues of spherical type, Buildings and the geometry of diagrams (Como, 1984), Lectures Notes in Math., vol. 1181, pp. 242–248. Springer, Berlin (1986) |
| [16] | Ronan, M.A.: Lectures on Buildings. Persp. Math. 7, Academic Press (1989) · Zbl 0694.51001 |
| [17] | Ronan M.A. and Tits J. (1987). Building Buildings. Math. Ann. 278: 291–306 · Zbl 0628.51001 · doi:10.1007/BF01458072 |
| [18] | Serre J.P. (1968). Corps locaux. Hermann, Paris · Zbl 0200.00002 |
| [19] | Tits, J.: Lecture Notes in Mathematics, vol. 386. Springer Verlag (1974) |
| [20] | Tits, J.: Buildings and group amalgamations. In: Proceedings of groups–St. Andrews 1985, pp. 110–127, London Math. Soc. Lecture Note Ser., vol. 121. Cambridge Univ. Press, Cambridge (1986) |
| [21] | Tits, J.: Sphere of Radius 2 in Triangle Buildings, I. Finite Geometries buildings and related Topics. Pingrie Park Conference (1988) |
| [22] | Tits, J., Weiss, R.M.: Moufang Polygons. Springer, (2002) · Zbl 1010.20017 |
| [23] | Van Maldeghem and H. (1990). Automorphisms of nonclassical triangle buildings. Bull. Soc. Math. Belg. Sér.B 42(2): 201–23 · Zbl 0780.51007 |
| [24] | Van Maldeghem and H. (1987). Nonclassical triangle buildings. Geom. Dedicata 24(2): 123–206 · Zbl 0648.51016 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
