zbMATH — the first resource for mathematics

Automorphisms of generalized Thompson groups. (English) Zbl 0930.20039
Richard J. Thompson introduced a triple of infinite groups \(F\subset T\subset G\) in the 1960s and showed that they have several interesting properties. These groups were later generalized to infinite groups \(F_{n,\infty}\subset F_n\subset T_{n,r}\subset G_{n,r}\) for \(n\geq 2\) and \(r\geq 1\) for which \(F=F_{2,\infty}=F_2\), \(T=T_{2,1}\) and \(G=G_{2,1}.\)
The automorphism groups \(\operatorname{Aut}(F)\) and \(\operatorname{Aut}(T)\) were analyzed by the first author [M. G. Brin, Publ. Math., Inst. Haut. Étud. Sci. 84, 5-33 (1996; Zbl 0891.57037)]. In this paper, the authors study \(\operatorname{Aut}(F_{n,\infty})\), \(\operatorname{Aut}(F_n)\), and \(\operatorname{Aut}(T_{n,n-1})\) for \(n>2\). These results differ sharply from the ones in [loc. cit.]. The automorphism groups of the generalizations are “large” and have “exotic” elements.

20F28 Automorphism groups of groups
20E36 Automorphisms of infinite groups
Full Text: DOI arXiv
[1] R. Bieri, R. Strebel, On groups of PL homeomorphisms of the real line, Math. Sem. der Univ. Frankfurt, Frankfurt am Main, 1985 · Zbl 1377.20002
[2] Brin, M.G., The chameleon groups of richard J. Thompson: automorphisms and dynamics, Inst. hautes études sci. publ. math., 84, 5-33, (1996) · Zbl 0891.57037
[3] M. G. Brin, The unbiquity of Thompson’s group F in groups of piecewise linear homeomorphisms of the unit interval, J. Lond. Math. Soc. · Zbl 0957.20025
[4] Brin, M.G.; Squier, C.C., Groups of piecewise linear homeomorphisms of the real line, Invent. math., 79, 485-498, (1985) · Zbl 0563.57022
[5] Brown, K.S., Finiteness properties of groups, J. pure appl. algebra, 44, 45-75, (1987) · Zbl 0613.20033
[6] Brown, K.S.; Geoghegan, R., An infinite-dimensional torsion-free FP_∞, Invent. math., 77, 367-381, (1984) · Zbl 0557.55009
[7] Cannon, J.W.; Floyd, W.J.; Parry, W.R., Introductory notes on richard Thompson’s groups, Enseign. math. (2), 42, 215-256, (1996) · Zbl 0880.20027
[8] Cohen, D.E., String rewriting—A survey for group theoreists, (), 37-47 · Zbl 0833.20037
[9] Cohn, P.M., Algebra, (1977), Wiley Chichester
[10] Dyer, J.L.; Formanek, E., The automorphism group of a free group is complete, J. London math. soc. (2), 11, 181-190, (1975) · Zbl 0313.20021
[11] Formanek, E., Characterizing a free group in its automorphism group, J. algebra, 133, 424-432, (1990) · Zbl 0715.20022
[12] Freyd, P.; Heller, A., Splitting homotopy idempotents, II, J. pure appl. algebra, 89, 93-106, (1993) · Zbl 0786.55008
[13] Gersten, S.M., Selected problems, (), 545-551 · Zbl 0616.20015
[14] Ghys, E.; Sergiescu, V., Sur un groupe remarquable de difféomorphismes du cercle, Comment. math. helv., 62, 185-239, (1987) · Zbl 0647.58009
[15] F. Gross, private communication
[16] V. S. Guba, M. V. Sapir, Diagram groups, University of Nebraska-Lincoln, 1996 · Zbl 0930.20033
[17] Higman, G., On infinite simple permutation groups, Publ. math. debrecen, 3, 221-226, (1954) · Zbl 0057.25801
[18] Higman, G., Finitely presented infinite simple groups, Notes on pure mathematics, 8, (1974), Australian National UniversityDepartment of Pure Mathematics Canberra
[19] W. Lück, Hilbert modules and modules over finite von Neumann algebras and applications toL2, Johannes Gutenberg-Universität Mainz, December 1995
[20] S. H. McCleary, M. Rubin, Locally moving groups and the reconstruction problem for chains and circles, Bowling Green State University, Bowling Green, Ohio, 1996
[21] Neumann, P.M., On the structure of standard wreath products of groups, Math. Z., 84, 343-373, (1964) · Zbl 0122.02901
[22] Plotkin, B.I., Groups of automorphisms of algebraic systems, (1972), Wolters-Noordhoff Groningen · Zbl 0229.20004
[23] Robinson, D.J.S., A course in the theory of groups, Graduate texts in math., 80, (1996), Springer-Verlag New York
[24] Stein, M., Groups of piecewise linear homeomorphisms, Trans. amer. math. soc., 332, 477-514, (1992) · Zbl 0798.20025
[25] R. J. Thompson, Handwritten, widely circulated, 1973+
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.