Open subgroups of the automorphism group of a right-angled building. (English) Zbl 1428.51004
Summary: We study the group of type-preserving automorphisms of a right-angled building, in particular when the building is locally finite. Our aim is to characterize the proper open subgroups as the finite index closed subgroups of the stabilizers of proper residues. One of the main tools is the new notion of firm elements in a right-angled Coxeter group, which are those elements for which the final letter in each reduced representation is the same. We also introduce the related notions of firmness for arbitrary elements of such a Coxeter group and \(n\)-flexibility of chambers in a right-angled building. These notions and their properties are used to determine the set of chambers fixed by the fixator of a ball. Our main result is obtained by combining these facts with ideas by Pierre-Emmanuel Caprace and Timothée Marquis in the context of Kac-Moody groups over finite fields, where we had to replace the notion of root groups by a new notion of root wing groups.
MSC:
| 51E24 | Buildings and the geometry of diagrams |
| 22D05 | General properties and structure of locally compact groups |
| 20F65 | Geometric group theory |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
| 20E42 | Groups with a \(BN\)-pair; buildings |
Keywords:
right-angled buildings; right-angled Coxeter groups; totally disconnected locally compact groups; open subgroupsReferences:
| [1] | Abramenko, P., Brown, K.S.: Buildings, Theory and Applications, volume 248 of Graduate Texts in Mathematics. Springer, New York (2008) · Zbl 1214.20033 |
| [2] | Burger, M., Mozes, S.: Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math. 92, 151-194 (2001) · Zbl 1007.22013 · doi:10.1007/BF02698916 |
| [3] | Baudisch, A., Martin-Pizarro, A., Ziegler, M.: A model-theoretic study of right-angled buildings. J. Eur. Math. Soc. (JEMS) 19(10), 3091-3141 (2017) · Zbl 1423.03120 · doi:10.4171/JEMS/736 |
| [4] | Bourdon, M.: Immeubles hyperboliques, dimension conforme et rigidité de Mostow. Geom. Funct. Anal. 7(2), 245-268 (1997) · Zbl 0876.53020 · doi:10.1007/PL00001619 |
| [5] | Caprace, P.-E.: Automorphism groups of right-angled buildings: simplicity and local splittings. Fund. Math. 224(1), 17-51 (2014) · Zbl 1296.20031 · doi:10.4064/fm224-1-2 |
| [6] | Caprace, P.-E., Marquis, T.: Open subgroups of locally compact Kac-Moody groups. Math. Z. 274(1-2), 291-313 (2013) · Zbl 1275.20057 · doi:10.1007/s00209-012-1070-4 |
| [7] | Caprace, P.-E., Rémy, B.: Simplicity and superrigidity of twin building lattices. Invent. Math. 176(1), 169-221 (2009) · Zbl 1173.22007 · doi:10.1007/s00222-008-0162-6 |
| [8] | Capdeboscq, I., Thomas, A.: Cocompact lattices in complete Kac-Moody groups with Weyl group right-angled or a free product of spherical special subgroups. Math. Res. Lett. 20(2), 339-358 (2013) · Zbl 1288.20064 · doi:10.4310/MRL.2013.v20.n2.a10 |
| [9] | Davis, M.W.: Buildings are \[{{\rm CAT}}(0)\] CAT(0). In: Geometry and Cohomology in Group Theory (Durham, 1994), volume 252 of London Math. Soc. Lecture Note Ser., pp. 108-123. Cambridge University Press, Cambridge (1998) · Zbl 0978.51005 |
| [10] | De Medts, T., Silva, A.C., Struyve, K.: Universal groups for right-angled buildings. Groups Geom. Dyn. 12, 231-288 (2018) · Zbl 1394.51005 · doi:10.4171/GGD/443 |
| [11] | Haglund, F., Paulin, F.: Constructions arborescentes d’immeubles. Math. Ann. 325(1), 137-164 (2003) · Zbl 1025.51014 · doi:10.1007/s00208-002-0373-x |
| [12] | Kubena, A., Thomas, A.: Density of commensurators for uniform lattices of right-angled buildings. J. Group Theory 15(5), 565-611 (2012) · Zbl 1276.20034 · doi:10.1515/jgt-2012-0017 |
| [13] | Macbeath, A.M., Świerczkowski, S.: On the set of generators of a subgroup. Nederl. Akad. Wetensch. Proc. Ser. A 62 = Indag. Math. 62(21), 280-281 (1959) · Zbl 0118.26503 |
| [14] | Reid, C.D.: Distal actions on coset spaces in totally disconnected, locally compact groups. ArXiv e-prints (2016) |
| [15] | Reid, C.D.: Dynamics of flat actions on totally disconnected, locally compact groups. New York J. Math. 22, 115-190 (2016) · Zbl 1332.22006 |
| [16] | Rémy, B., Ronan, M.: Topological groups of Kac-Moody type, right-angled twinnings and their lattices. Comment. Math. Helv. 81(1), 191-219 (2006) · Zbl 1163.22009 · doi:10.4171/CMH/49 |
| [17] | Thomas, A.: Lattices acting on right-angled buildings. Algebr. Geom. Topol. 6, 1215-1238 (2006) · Zbl 1128.22002 · doi:10.2140/agt.2006.6.1215 |
| [18] | Tits, J.: Le problème des mots dans les groupes de Coxeter. In: Symposia Mathematica (INDAM. Rome, 1967/68), vol. 1, pp. 175-185. Academic Press, London (1969) · Zbl 0206.03002 |
| [19] | Thomas, A., Wortman, K.: Infinite generation of non-cocompact lattices on right-angled buildings. Algebr. Geom. Topol. 11(2), 929-938 (2011) · Zbl 1266.20043 · doi:10.2140/agt.2011.11.929 |
| [20] | Weiss, R.M.: The Structure of Affine Buildings, volume 168 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ (2009) · Zbl 1166.51001 |
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