Hostname: page-component-7dd5485656-c8zdz Total loading time: 0 Render date: 2025-10-30T04:38:41.223Z Has data issue: false hasContentIssue false
  • English
  • Français

Groups acting simply transitively on vertex sets of hyperbolic triangular buildings

Part of: Special aspects of infinite or finite groups Algebraic combinatorics

Published online by Cambridge University Press:  01 May 2012

Lisa Carbone ,
Riikka Kangaslampi  and
Alina Vdovina
Lisa Carbone
Affiliation:
Department of Mathematics, Rutgers, The State University of New Jersey, Hill Center, Busch Campus, 110 Frelinghuysen Rd, Piscataway, NJ 08854, USA (email: carbonel@math.rutgers.edu)
Riikka Kangaslampi
Affiliation:
Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076 Aalto, Finland (email: riikka.kangaslampi@aalto.fi)
Alina Vdovina
Affiliation:
School of Mathematics and Statistics, University of Newcastle, Newcastle upon Tyne, NE1 7RU, United Kingdom (email: alina.vdovina@ncl.ac.uk)

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct and classify all groups given by triangular presentations associated to the smallest thick generalized quadrangle that act simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial thickness. Our classification yields 23 non-isomorphic torsion-free groups (which were obtained in an earlier work) and 168 non-isomorphic torsion groups acting on one of two possible buildings with the smallest thick generalized quadrangle as the link of each vertex. In analogy with the case, we find both torsion and torsion-free groups acting on the same building.

MSC classification

Secondary: 05E15: Combinatorial aspects of groups and algebras 20F65: Geometric group theory 20F67: Hyperbolic groups and nonpositively curved groups 05E18: Group actions on combinatorial structures

Information

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 101 - 112
Copyright
Copyright © London Mathematical Society 2012

References

[1]Ballmann, W. and Brin, M., ‘Polygonal complexes and combinatorial group theory’, Geom. Dedicata 50 (1994) 165–191.CrossRefGoogle Scholar
[2]Bourdon, M., ‘Immeubles hyperboliques, dimension conforme et rigidité de Mostow’, Geom. Funct. Anal. 7 (1997) 245–268.CrossRefGoogle Scholar
[3]Bourdon, M., ‘Sur les immeubles fuchsiens et leur type de quasi-isométrie (French) [Fuchsian buildings and their quasi-isometry type]’, Ergodic Theory Dynam. Systems 20 (2000) 343–364.CrossRefGoogle Scholar
[4]Capdeboscq, I. (Korchagina) and Thomas, A., ‘Lattices in complete rank 2 Kac-Moody groups’, J. Pure Appl. Algebra 216 (2012) 1338–1371.CrossRefGoogle Scholar
[5]Carbone, L., Cartwright, D. and Steger, T., Cocompact lattices in hyperbolic Kac-Moody groups, Preprint, 2006.Google Scholar
[6]Carbone, L. and Cobbs, C., ‘Infinite descending chains of cocompact lattices in Kac-Moody groups’, J. Algebra Appl. 10 (2011) 1187–1219.CrossRefGoogle Scholar
[7]Carbone, L. and Garland, H., ‘Existence of lattices in Kac-Moody groups over finite fields’, Commun. Contemp. Math. 5 (2003) 813–867.CrossRefGoogle Scholar
[8]Cartwright, D., Mantero, A., Steger, T. and Zappa, A., ‘Groups acting simply transitively on vertices of a building of type ’, Geom. Dedicata 47 (1993) 143–166.CrossRefGoogle Scholar
[9]Cartwright, D., Mantero, A., Steger, T. and Zappa, A., ‘Groups acting simply transitively on vertices of a building of type , II: the cases q=2 and q=3’, Geom. Dedicata 47 (1993) 167–223.CrossRefGoogle Scholar
[10]Cartwright, D. and Steger, T., ‘Enumeration of the 50 fake projective planes’, C. R. Math. Acad. Sci. Paris 348 (2010) 11–13.CrossRefGoogle Scholar
[11]Edjvet, M. and Howie, J., ‘Star graphs, projective planes and free subgroups in small cancellation groups’, Proc. Lond. Math. Soc. (3) 57 (1988) 301–328.CrossRefGoogle Scholar
[12]Essert, J., ‘A geometric construction of panel-regular lattices in buildings of types and ’, Preprint, 2010, arXiv:0908.2713v3.Google Scholar
[13]Gaboriau, D. and Paulin, F., ‘Sur les immeubles hyperboliques’, Geom. Dedicata 88 (2001) 153–197.CrossRefGoogle Scholar
[14]Ghys, E. and de la Harpe (eds), P., Sur les groupes hyperboliques d’après Mikhael Gromov (Birkhäuser, Boston, 1990).CrossRefGoogle Scholar
[15]Haglund, F., ‘Existence, uniqueness and homogeneity of certain hyperbolic buildings’, Math. Z. 242 (2002) 97–148.CrossRefGoogle Scholar
[16]Kangaslampi, R. and Vdovina, A., ‘Cocompact actions on hyperbolic buildings’, Internat. J. Algebra Comput. 20 (2010) 591–603.CrossRefGoogle Scholar
[17]Kantor, W. M., Liebler, R. A. . and Tits, J., ‘On discrete chamber-transitive automorphism groups of affine buildings’, Bull. Amer. Math. Soc. (N.S.) 16 (1987) 129–133.CrossRefGoogle Scholar
[18]Kato, F. and Ochiai, H., ‘Arithmetic structure of CMSZ fake projective planes’, J. Algebra 305 (2006) 116–1185.CrossRefGoogle Scholar
[19]Rémy, B., ‘Groupes de Kac-Moody déployés et presque déployés (French) [Split and almost split Kac-Moody groups]’, Astérisque 277 (2002) 1–348.Google Scholar
[20]Rémy, B. and Ronan, M., ‘Topological groups of Kac-Moody type, right-angled twinnings and their lattices’, Comment Math. Helv. 81 (2006) 191–219.CrossRefGoogle Scholar
[21]Swiatkowski, J., ‘Trivalent polygonal complexes of nonpositive curvature and Platonic symmetry’, Geom. Dedicata 70 (1998) 87–110.CrossRefGoogle Scholar
[22]Tits, J. and Weiss, R. M., Moufang polygons, Springer Monographs in Mathematics (Springer, Berlin, 2002).CrossRefGoogle Scholar
[23]Vdovina, A., ‘Combinatorial structure of some hyperbolic buildings’, Math. Z. 241 (2002) 471–478.CrossRefGoogle Scholar
[24]Wise, D., ‘The residual finiteness of negatively curved polygons of finite groups’, Invent. Math. 149 (2002) 579–617.CrossRefGoogle Scholar
[25]Xie, X., ‘Quasi-isometric rigidity of Fuchsian buildings’, Topology 45 (2006) 101–169.CrossRefGoogle Scholar