A radius $1$ irreducibility criterion for lattices in products of trees

Caprace, Pierre-Emmanuel

doi:10.5802/ahl.132

Caprace, Pierre-Emmanuel

A radius

1

irreducibility criterion for lattices in products of trees

Annales Henri Lebesgue, Volume 5 (2022), pp. 643-675.

Metadata

DOI10.5802/ahl.132

Keywords Irreducible lattice, groups acting on trees, just-infinite group, $2$-transitive group

Abstract

Let $T_{1}, T_{2}$ be regular trees of degrees $d_{1}, d_{2} \geq 3$ . Let also $Γ \leq Aut (T_{1}) \times Aut (T_{2})$ be a group acting freely and transitively on $V T_{1} \times V T_{2}$ . For $i = 1$ and $2$ , assume that the local action of $Γ$ on $T_{i}$ is $2$ -transitive; if moreover $d_{i} \geq 7$ , assume that the local action contains $Alt (d_{i})$ . We show that $Γ$ is irreducible, unless $(d_{1}, d_{2})$ belongs to an explicit small set of exceptional values. This yields an irreducibility criterion for $Γ$ that can be checked purely in terms of its local action on a ball of radius $1$ in $T_{1}$ and $T_{2}$ . Under the same hypotheses, we show moreover that if $Γ$ is irreducible, then it is hereditarily just-infinite, provided the local action on $T_{i}$ is not the affine group $F_{5} ⋊ F_{5}^{*}$ . The proof of irreducibility relies, in several ways, on the Classification of the Finite Simple Groups.

References

[Bas93] Bass, Hyman Covering theory for graphs of groups, J. Pure Appl. Algebra, Volume 89 (1993) no. 1-2, pp. 3-47 | DOI | MR | Zbl

[Bau07] Baumeister, Barbara Primitive permutation groups with a regular subgroup, J. Algebra, Volume 310 (2007) no. 2, pp. 569-618 | DOI | MR | Zbl

[BL01] Bass, Hyman; Lubotzky, Alexander Tree lattices, Progress in Mathematics, 176, Birkhäuser, 2001 (with appendices by H. Bass, L. Carbone, A. Lubotzky, G. Rosenberg and J. Tits) | DOI | MR | Zbl

[BM97] Burger, Marc; Mozes, Shahar Finitely presented simple groups and products of trees, C. R. Math. Acad. Sci. Paris, Volume 324 (1997) no. 7, pp. 747-752 | DOI | MR | Zbl

[BM00a] Burger, Marc; Mozes, Shahar Groups acting on trees: from local to global structure, Publ. Math., Inst. Hautes Étud. Sci. (2000) no. 92, p. 113-150 (2001) | DOI | Numdam | MR | Zbl

[BM00b] Burger, Marc; Mozes, Shahar Lattices in product of trees, Publ. Math., Inst. Hautes Étud. Sci. (2000) no. 92, pp. 151-194 | DOI | Numdam | MR | Zbl

[BS06] Bader, Uri; Shalom, Yehuda Factor and normal subgroup theorems for lattices in products of groups, Invent. Math., Volume 163 (2006) no. 2, pp. 415-454 | DOI | MR | Zbl

[Cam99] Cameron, Peter J. Permutation groups, London Mathematical Society Student Texts, 45, Cambridge University Press, 1999 | DOI | MR | Zbl

[Cap19] Caprace, Pierre-Emmanuel Finite and infinite quotients of discrete and indiscrete groups, Groups St Andrews 2017 in Birmingham (London Mathematical Society Lecture Note Series), Volume 455, Cambridge University Press, 2019, pp. 16-69 | DOI | MR

[CDM13] Caprace, Pierre-Emmanuel; De Medts, Tom Trees, contraction groups, and Moufang sets, Duke Math. J., Volume 162 (2013) no. 13, pp. 2413-2449 | DOI | MR | Zbl

[Con09] Conder, Marston On symmetries of Cayley graphs and the graphs underlying regular maps, J. Algebra, Volume 321 (2009) no. 11, pp. 3112-3127 | DOI | MR | Zbl

[Gor82] Gorenstein, Daniel Finite simple groups, University Series in Mathematics, Plenum Press, 1982 (An introduction to their classification) | DOI | MR | Zbl

[HL00] Huppert, Bertram; Lempken, Wolfgang Simple groups of order divisible by at most four primes, Izv. Gomel. Gos. Univ. Im. F. Skoriny, Volume 2000 (2000) no. 3(16), pp. 64-75 | Zbl

[KS04] Kurzweil, Hans; Stellmacher, Bernd The theory of finite groups. An introduction, Universitext, Springer, 2004 (translated from the 1998 German original) | DOI | MR | Zbl

[Li05] Li, Cai Heng Finite $s$ -arc transitive Cayley graphs and flag-transitive projective planes, Proc. Am. Math. Soc., Volume 133 (2005) no. 1, pp. 31-41 | DOI | MR | Zbl

[LL09] Li, Jing Jian; Lu, Zai Ping Cubic $s$ -arc transitive Cayley graphs, Discrete Math., Volume 309 (2009) no. 20, pp. 6014-6025 | DOI | MR | Zbl

[LL16] Ling, Bo; Lou, Ben Gong A 2-arc transitive pentavalent Cayley graph of $A_{39}$ , Bull. Aust. Math. Soc., Volume 93 (2016) no. 3, pp. 441-446 | DOI | MR | Zbl

[LL17] Ling, Bo; Lou, Ben Gong Arc-transitive pentavalent Cayley graphs with soluble vertex stabilizer on finite nonabelian simple groups (2017) (https://arxiv.org/abs/1702.05754)

[LPS00] Liebeck, Martin W.; Praeger, Cheryl E.; Saxl, Jan Transitive subgroups of primitive permutation groups, J. Algebra, Volume 234 (2000) no. 2, pp. 291-361 (Special issue in honor of Helmut Wielandt) | DOI | MR | Zbl

[LPS10] Liebeck, Martin W.; Praeger, Cheryl E.; Saxl, Jan Regular subgroups of primitive permutation groups, Mem. Am. Math. Soc., Volume 203 (2010) no. 952, vi | DOI | MR | Zbl

[LX14] Li, Cai Heng; Xia, Binzhou Factorizations of almost simple groups with a solvable factor, and Cayley graphs of solvable groups (2014) (https://arxiv.org/abs/1408.0350v1)

[Neb00] Nebbia, Claudio Minimally almost periodic totally disconnected groups, Proc. Am. Math. Soc., Volume 128 (2000) no. 2, pp. 347-351 | DOI | MR | Zbl

[Rad17] Radu, Nicolas New simple lattices in products of trees and their projections (2017) (https://arxiv.org/abs/1712.01091)

[Rad18] Radu, Nicolas Simple groups and lattices in trees and buildings: constructions and classifications, Ph. D. Thesis, Université catholique de Louvain, Belgique (2018)

[Rat04] Rattaggi, Diego Attilio Computations in groups acting on a product of trees: Normal subgroup structures and quaternion lattices, Ph. D. Thesis, ETH Zürich (Switzerland) (2004)

[Rob96] Robinson, Derek J. S. A course in the theory of groups, Graduate Texts in Mathematics, 80, Springer, 1996 | DOI | MR | Zbl

[Ser77] Serre, Jean-Pierre Arbres, amalgames, ${SL}_{2}$ , Astérisque, 46, Société Mathématique de France, 1977 (avec un sommaire anglais, rédigé avec la collaboration de Hyman Bass) | MR | Zbl

[Tro07] Trofimov, Vladimir I. Vertex stabilizers of graphs and tracks. I, Eur. J. Comb., Volume 28 (2007) no. 2, pp. 613-640 | DOI | MR | Zbl

[TW95] Trofimov, Vladimir I.; Weiss, Richard M. Graphs with a locally linear group of automorphisms, Math. Proc. Camb. Philos. Soc., Volume 118 (1995) no. 2, pp. 191-206 | DOI | MR | Zbl

[Wei79] Weiss, Richard M. Groups with a $(B, N)$ -pair and locally transitive graphs, Nagoya Math. J., Volume 74 (1979), pp. 1-21 | DOI | MR | Zbl

[Wis96] Wise, Daniel T. Non-positively curved squared complexes: Aperiodic tilings and non-residually finite groups, Ph. D. Thesis, Princeton University, Princeton, USA (1996) (https://www.proquest.com/docview/304259249, 126 pages)

[WW80] Wiegold, James; Williamson, Alan G. The factorisation of the alternating and symmetric groups, Math. Z., Volume 175 (1980) no. 2, pp. 171-179 | DOI | MR | Zbl

[XFWX05] Xu, Shang Jin; Fang, Xin Gui; Wang, Jie; Xu, Ming Yao On cubic $s$ -arc transitive Cayley graphs of finite simple groups, Eur. J. Comb., Volume 26 (2005) no. 1, pp. 133-143 | DOI | MR | Zbl

Metadata

Abstract

References

Cited by