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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.

##### MSC:
 20F28 Automorphism groups of groups 20E36 Automorphisms of infinite groups
##### Keywords:
generalized Thompson groups; automorphism groups
Full Text:
##### References:
 [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+
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