×

zbMATH — the first resource for mathematics

The Chameleon groups of Richard J. Thompson: Automorphisms and dynamics. (English) Zbl 0891.57037
Let \(PL_2(\mathbf R)\) denote the set of homeomorphisms \(f: \mathbf R \longrightarrow \mathbf R\) satisfying:   (1) \(f\) is piecewise linear;   (2) \(f\) is orientation preserving;   (3) all slopes of \(f\) are integral powers of 2;   (4) the “breaks” of \(f\) (discontinuities of \(f'\)) are in a discrete subset of \(\mathbf Z[\frac{1}{2}]\); and   (5) \(f(\mathbf Z[\frac{1}{2}])\subseteq \mathbf Z[\frac{1}{2}]\). Let \(BPL_2(\mathbf R)\) be the elements \(f\) of \(PL_2(\mathbf R)\) whose support (i.e., the set of points \(x\) such that \(f(x)\neq x\)) is a bounded subset of \(\mathbf R\). Let \(F\) be those elements \(f\) of \(PL_2(\mathbf R)\) that are translations by integers near \(\pm \infty\) in the sense that there are integers \(i\) and \(j\) and a real \(M\) so that \(f(x)=x+i\) for all \(x>|M|\) and \(f(x)=x+j\) for all \(x<-|M|\). One has \(BPL_2(\mathbf R)\subseteq F\). Let \(T\) be those homeomorphisms from \(S^1=\mathbf R/\mathbf Z\) to itself that satisfy (1)–(5) above. Denote by \(\widetilde{PL}_2(\mathbf R)\) and \(\widetilde T\) the groups obtained if in the definitions of \(PL_2(\mathbf R)\) and \(T\) respectively, (2) is replaced by: (2’) \(f\) is orientation preserving or orientation reversing.
The main result of this paper is the following theorem: Let \(G\) be a group for which \(BPL_2(\mathbf R)\subseteq G \subseteq \widetilde{PL}_2(\mathbf R)\) or \(T\subseteq G \subseteq \widetilde T\). Let \(N(G)\) be the normalizer of \(G\) in \(\text{Homeo}(\mathbf R)\) or \(\text{Homeo}(S^1)\) respectively. Then (i) the natural homomorphism \(N(G)\longrightarrow \text{Aut}(G)\) is an isomorphism, (ii) \(N(G)\subseteq \widetilde {PL}_2(\mathbf R)\) or \(N(G)\subseteq \widetilde T\) whichever applies, (iii) the containment in (ii) is equality whenever \(G\) is one of \(BPL_2(\mathbf R)\), \(PL_2(\mathbf R)\), \(\widetilde {PL}_2(\mathbf R)\), \(T\), \(\widetilde T\), and (iv) if \(N_+(F)\) represents the index 2 subgroup of \(N(F)\) of orientation preserving elements, then there is a short exact sequence \[ 1\longrightarrow F\longrightarrow N_+(F)\longrightarrow T\times T \longrightarrow 1. \]
Reviewer: V.L.Popov (Moskva)

MSC:
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
20F99 Special aspects of infinite or finite groups
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML arXiv
References:
[1] R. Bieri andR. Strebel,On groups of PL-homeomorphisms of the real line, preprint, Math. Sem. der Univ. Frankfurt, Frankfurt, 1985. · Zbl 1377.20002
[2] M. G. Brin andC. C. Squier, Groups of piecewise linear homeomorphisms of the real line,Invent. Math.,79 (1985), 485–498. · Zbl 0563.57022 · doi:10.1007/BF01388519
[3] K. S. Brown, Finiteness properties of groups,J. Pure and Applied Algebra,44 (1987), 45–75. · Zbl 0613.20033 · doi:10.1016/0022-4049(87)90015-6
[4] K. S. Brown, The geometry of finitely presented infinite simple groups,Algorithms and Classification in Combinatorial Group Theory, G. Baumslag and C. F. Miller, III, Eds., MSRI Publications, Number 23, Springer-Verlag, New York, 1991, p. 121–136.
[5] K. S. Brown, The geometry of rewriting systems: a proof of the Anick-Groves-Squier Theorem,, p. 137–163. · Zbl 0764.20016
[6] K. S. Brown andR. Geoghegan, An infinite-dimensional torsion-free FPgroup,Invent. Math.,77 (1984), 367–381. · Zbl 0557.55009 · doi:10.1007/BF01388451
[7] J. W. Cannon, W. J. Floyd andW. R. Parry, Notes on Richard Thompson’s groups F and T, to appear inL’Enseignement Mathématique. · Zbl 0880.20027
[8] C. G. Chehata, An algebraically simple ordered group,Proc. Lond. Math. Soc. (3),2 (1952), 183–197. · Zbl 0046.02501 · doi:10.1112/plms/s3-2.1.183
[9] S. Cleary, Groups of piecewise-linear homeomorphisms with irrational slopes,Rocky Mountain J. Math.,25 (1995), 935–955. · Zbl 0857.57023 · doi:10.1216/rmjm/1181072197
[10] J. Dydak andJ. Segal, Shape Theory: An Introduction,Lecture Notes in Math., Number 688, Springer-Verlag, Berlin, 1978. · Zbl 0401.54028
[11] P. Freyd andA. Heller, Splitting homotopy idempotents, II,J. Pure and Applied Algebra,89 (1993), 93–106. · Zbl 0786.55008 · doi:10.1016/0022-4049(93)90088-B
[12] E. Ghys andV. Sergiescu,Sur un groupe remarquable de difféomorphismes du cercle, preprint IHES/M/85/65, Institut des Hautes Études Scientifiques, Bures-sur-Yvette, 1985. · Zbl 0647.58009
[13] E. Ghys andV. Sergiescu, Sur un groupe remarquable de difféomorphismes du cercle,Comment. Math. Helvetici,62 (1987), 185–239. · Zbl 0647.58009 · doi:10.1007/BF02564445
[14] P. Greenberg, Pseudogroups from group actions,Amer. J. Math.,109 (1987), 893–906. · Zbl 0644.57012 · doi:10.2307/2374493
[15] P. Greenberg, Projective aspects of the Higman-Thompson group,Group Theory from a Geometrical Viewpoint, ICTP conference, Triest, Italy, 1990, E. Ghys, A. Haefliger, A. Verjovsky, Editors, World Scientific, Singapore, 1991, p. 633–644. · Zbl 0860.57038
[16] P. Greenberg andV. Sergiescu, An acyclic extension of the braid group,Comm. Math. Helvetici,66 (1991), 109–138. · Zbl 0736.20020 · doi:10.1007/BF02566638
[17] H. M. Hastings andA. Heller, Homotopy idempotents on finite-dimensional complexes split,Proc. Amer. Math. Soc.,85 (1982), 619–622. · Zbl 0513.55011 · doi:10.1090/S0002-9939-1982-0660617-5
[18] S. H. McCleary, Groups of homeomorphisms with manageable automorphism groups,Comm. in Algebra,6 (1978), 497–528. · Zbl 0377.20035 · doi:10.1080/00927877808822256
[19] S. H. McCleary andM. Rubin,Locally moving groups and the reconstruction problem for chains and circles, preprint, Bowling Green State University, Bowling Green, Ohio.
[20] R. McKenzie andR. J. Thompson,An elementary construction of unsolvable word problems in group theory, Word Problems, Boone, Cannonito and Lyndon Eds., North Holland, 1973, p. 457–478. · Zbl 0286.02047
[21] J. N. Mather, Integrability in codimension 1,Comment. Math. Helvetici,48 (1973), 195–233. · Zbl 0284.57016 · doi:10.1007/BF02566122
[22] E. A. Soott, A tour around finitely presented infinite simple groups,Algorithms and Classification in Combinatorial Group Theory, G. Baumslag and C. F. Miller, III, Eds., MSRI Publications, Number 23, Springer-Verlag, New York, 1991, p. 83–119.
[23] M. Stein, Groups of piecewise linear homeomorphisms,Trans. Amer. Math. Soc.,332 (1992), 477–514. · Zbl 0798.20025 · doi:10.2307/2154179
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.