Duke Mathematical Journal

Trees, contraction groups, and Moufang sets

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 $\operatorname {PSL}_{2}(k)\leq G\leq\operatorname {PGL}_{2}(k)$. 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 $\partial T$ 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

https://projecteuclid.org/euclid.dmj/1381238849

Digital Object Identifier
doi:10.1215/00127094-2371640

Mathematical Reviews number (MathSciNet)
MR3127805

Zentralblatt MATH identifier
1291.20025

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

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.