Abstract
We show that the abstract commensurator of a nearly level transitive weakly branch group H coincides with the relative commensurator of H in the homeomorphism group of the boundary of the tree on which H acts. It is also shown that the commensurator of an infinite group which is commensurable with its own nth direct power \((2 \leqslant n \in \mathbb{N})\) contains a Higman–Thompson group as a subgroup. Applying these results to ‘the’ Grigorchuk 2-group G we show that the commensurator of G is a finitely presented infinite simple group.
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[AB94] A'Campo, N. and Burger, M.: Réseaux arithmétiques et commensurateur d'aprés G. A. Margulis, Invent. Math. 166 (1994), 1–25.
[BdlH97] Burger, M. and de la Harpe, P.: Constructing irreducible representations of discrete groups, Proc. Indian Acad. Sci. Math. Sci. 107 (1997), 223–235.
[BS85] Brin, M. G. and Squier, C. C.: Groups of piecewise linear homeomorphisms of the real line, Invent. Math. 79 (1985), 485–498.
[BSV99] Brunner, A., Sidki, S. and Vieira, A.: A just non-solvable torsion free group defined on the binary tree, J. Algebra 211 (1999), 99–114.
[CFP96] Cannon, J. W., Floyd, W. J. and Perry, W. R.: Introductory notes on Richard Thompson's group, Enseign. Math. (2) 42 (1996), 215–256.
[dlH00] de la Harpe, P.: Topics in Geometric Group Theory, University of Chicago Press, 2000.
[Gri80] Grigorchuk, R. I.: On Burnside's problem for periodic groups, Funct. Anal. Appl. 14 (1980), 41–43.
[Gri85] Grigorchuk, R. I.: Degrees of growth of finitely generated groups and the theory of invariant means, Math. USSR Izv. 25 (1985).
[Gri98] Grigorchuk, R. I.: A finitely presented amenable group not in the class EG, Sb. Math. 189 (1998), 75–95.
[Gri00] Grigorchuk, R. I.: Just infinite branch groups, In: M. du Sautoy, D. Segal, and A. Shalev (eds), New Horizons in Pro-p Groups, Birkhäuser, Boston, 2000.
[GS83] Gupta, N. and Sidki, S.: On the Burnside problem for periodic groups, Math. Z. 182 (1983), 385–388.
[GS97] Guba, V. and Sapir, M.: Diagram groups, Mem. Amer. Math. Soc. 620 (1997).
[Hig74] Higman, G.: Finitely presented in.nite simple groups, Notes Pure Math. 8, The Australian National University, 1974.
[LN02] Lavreniuk, Y. and Nekrashevych, V.: Rigidity of branch groups acting on rooted trees, Geom. Dedicata 89 (2002), 159–179.
[Lys85] Lysionok, I. G.: A system of defining relations for the Grigorchuk group, Math. Notes 38 (1985), 784–792. Russian original: pp. 503–516.
[Neu86] Neumann, P. M.: Some questions of Edjevet and Pride about infinite groups, Illinois J. Math. 30 (1986), 301–316.
[Röv99a] Röver, C. E.: Constructing finitely presented simple groups that contain Grigorchuk groups, J. Algebra 220 (1999), 284–313.
[Röv99b] Röer, C. E.: Subgroups of finitely presented simple groups, DPhil Thesis, Oxford, 1999.
[Roz93] Rozhkov, A. V.: Centralizers of elements in a group of tree automorphisms, Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), 82–105.
[Sco84] Scott, E. A.: A construction which can be used to produce finitely presented simple groups, J. Algebra 90 (1984), 294–322.
[Seg01] Segal, D.: The finite images of finitely generated groups, Proc. London Math. Soc. 82 (3) (2001), 597–613.
[Ste92] Stein, M.: Groups of piecewise linear homeomorphisms, Trans. Amer. Math. Soc. 332 (1992), 477–514.
[Tho80] Thompson, R. J.: Embeddings into finitely generated simple groups which preserve the word problem, In: S. I. Adian, W. W. Boone, and G. Higman (eds), Word Problems II, North-Holland, Amsterdam, 1980.
[TJ74] Tyrer Jones, J. M.: Direct products and the Hopf property, J. Austral. Math. Soc. 17 (1974), 174–196.
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Röver, C.E. Abstract Commensurators of Groups Acting on Rooted Trees. Geometriae Dedicata 94, 45–61 (2002). https://doi.org/10.1023/A:1020916928393
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DOI: https://doi.org/10.1023/A:1020916928393