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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
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