Duke Mathematical Journal

Trees, contraction groups, and Moufang sets

Pierre-Emmanuel Caprace and Tom De Medts

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study closed subgroups G of the automorphism group of a locally finite tree T acting doubly transitively on the boundary. We show that if the stabilizer of some end is metabelian, then there is a local field k such that PSL2(k)GPGL2(k). We also show that the contraction group of some hyperbolic element is closed and torsion-free if and only if G is (virtually) a rank 1 simple p-adic analytic group for some prime p. A key point is that if some contraction group is closed, then G is boundary-Moufang, meaning that the boundary T is a Moufang set. We collect basic results on Moufang sets arising at infinity of locally finite trees, and we provide a complete classification in case the root groups are torsion-free.

Article information

Source
Duke Math. J., Volume 162, Number 13 (2013), 2413-2449.

Dates
First available in Project Euclid: 8 October 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1381238849

Digital Object Identifier
doi:10.1215/00127094-2371640

Mathematical Reviews number (MathSciNet)
MR3127805

Zentralblatt MATH identifier
1291.20025

Subjects
Primary: 20E08: Groups acting on trees [See also 20F65]
Secondary: 20E42: Groups with a $BN$-pair; buildings [See also 51E24] 20G25: Linear algebraic groups over local fields and their integers 22D05: General properties and structure of locally compact groups

Citation

Caprace, Pierre-Emmanuel; De Medts, Tom. Trees, contraction groups, and Moufang sets. Duke Math. J. 162 (2013), no. 13, 2413--2449. doi:10.1215/00127094-2371640. https://projecteuclid.org/euclid.dmj/1381238849


Export citation

References

  • [1] Y. Barnea, M. Ershov, and T. Weigel, Abstract commensurators of profinite groups, Trans. Amer. Math. Soc. 363 (2011), no. 10, 5381–5417.
  • [2] U. Baumgartner and G. A. Willis, Contraction groups and scales of automorphisms of totally disconnected locally compact groups, Israel J. Math. 142 (2004), 221–248.
  • [3] A. Borel and J. Tits, Homomorphismes “abstraits” de groupes algébriques simples, Ann. of Math. (2) 97 (1973), 499–571.
  • [4] M. Burger and S. Mozes, $CAT(-1)$-spaces, divergence groups and their commensurators, J. Amer. Math. Soc. 9 (1996), 57–93.
  • [5] M. Burger and S. Mozes, Groups acting on trees: From local to global structure, Publ. Math. Inst. Hautes Études Sci. 92 (2000), 113–150 (2001).
  • [6] P.-E. Caprace, Y. de Cornulier, N. Monod, and R. Tessera, Amenable hyperbolic groups, to appear in J. Eur. Math. Soc., preprint, arXiv:1202.3585v1 [math.GR].
  • [7] J. Cossey and S. Stonehewer, On the derived length of finite dinilpotent groups, Bull. Lond. Math. Soc. 30 (1998), 247–250.
  • [8] T. De Medts and Y. Segev, A course on Moufang sets, Innov. Incidence Geom. 9 (2009), 79–122.
  • [9] T. De Medts, Y. Segev, and K. Tent, Special Moufang sets, their root groups and their $\mu$-maps, Proc. Lond. Math. Soc. (3) 96 (2008), 767–791.
  • [10] T. De Medts and R. M. Weiss, Moufang sets and Jordan division algebras, Math. Ann. 335 (2006), 415–433.
  • [11] J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic Pro-$p$ Groups, 2nd ed., Cambridge Stud. Adv. Math. 61, Cambridge Univ. Press, Cambridge, 1999.
  • [12] H. Glöckner and G. A. Willis, Classification of the simple factors appearing in composition series of totally disconnected contraction groups, J. Reine Angew. Math. 643 (2010), 141–169.
  • [13] M. Grüninger, Special Moufang sets with abelian Hua subgroup, J. Algebra 323 (2010), 1797–1801.
  • [14] W. Hazod and E. Siebert, Continuous automorphism groups on a locally compact group contracting modulo a compact subgroup and applications to stable convolution semigroups, Semigroup Forum 33 (1986), 111–143.
  • [15] N. Itô, Über das Produkt von zwei abelschen Gruppen, Math. Z. 62 (1955), 400–401.
  • [16] Gy. Károlyi, S. J. Kovács, and P. P. Pálfy, Doubly transitive permutation groups with abelian stabilizers, Aequationes Math. 39 (1990), 161–166.
  • [17] O. H. Kegel, Produkte nilpotenter Gruppen, Arch. Math. (Basel) 12 (1961), 90–93.
  • [18] O. H. Kegel, On the solvability of some factorized linear groups, Illinois J. Math. 9 (1965), 535–547.
  • [19] V. D. Mazurov, Doubly transitive permutation groups (in Russian), Sibirsk. Mat. Zh. 31, no. 4 (1990), 102–104, 222; English translation in Sib. Math. J. 31 (1990), 615–617.
  • [20] E. A. Pennington, On products of finite nilpotent groups, Math. Z. 134 (1973), 81–83.
  • [21] V. P. Platonov, The problem of strong approximation and the Kneser-Tits hypothesis for algebraic groups (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 1211–1219.
  • [22] Y. Segev, Proper Moufang sets with abelian root groups are special, J. Amer. Math. Soc. 22 (2009), 889–908.
  • [23] Y. Segev and R. M. Weiss, On the action of the Hua subgroups in special Moufang sets, Math. Proc. Cambridge Philos. Soc. 144 (2008), 77–84.
  • [24] J.-P. Serre, Lie Algebras and Lie Groups, Lectures given at Harvard University, vol. 1964, W. A. Benjamin, New York, 1965.
  • [25] F. G. Timmesfeld, Abstract Root Subgroups and Simple Groups of Lie Type, Monogr. Math. 95, Birkhäuser, Basel, 2001.
  • [26] J. Tits, Généralisations des groupes projectifs basées sur leurs propriétés de transitivité, Acad. Roy. Belgique. Cl. Sci. Mém. Coll. in $8^{\circ}$ 27 (1952), no. 2, 115.
  • [27] J. Tits, Algebraic and abstract simple groups, Ann. of Math. (2) 80 (1964), 313–329.
  • [28] J. Tits, “Classification of algebraic semisimple groups” in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, 1966, 33–62.
  • [29] J. Tits, “Sur le groupe des automorphismes d’un arbre” in Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, 188–211.
  • [30] J. Tits, “Twin buildings and groups of Kac-Moody type” in Groups, Combinatorics & Geometry (Durham, 1990), London Math. Soc. Lecture Note Ser. 165, Cambridge Univ. Press, Cambridge, 1992, 249–286.
  • [31] J. Tits and R. M. Weiss, Moufang Polygons, Springer Monogr. Math., Springer, Berlin, 2002.
  • [32] B. Ju. Veĭsfeĭler, The classification of semi-simple Lie algebras over ${\mathfrak{p}}$-adic field (in Russian), Dokl. Akad. Nauk SSSR 158 (1964), 258–260.
  • [33] J. S. P. Wang, The Mautner phenomenon for $p$-adic Lie groups, Math. Z. 185 (1984), 403–412.
  • [34] H. Wielandt, Über Produkte von nilpotenten Gruppen, Illinois J. Math. 2 (1958), 611–618.
  • [35] G. A. Willis, The structure of totally disconnected, locally compact groups, Math. Ann. 300 (1994), 341–363.
  • [36] G. A. Willis, Compact open subgroups in simple totally disconnected groups, J. Algebra 312 (2007), 405–417.