By Ian Chiswell

ISBN-10: 1107024811

ISBN-13: 9781107024816

The speculation of R-trees is a well-established and significant region of geometric team thought and during this publication the authors introduce a building that gives a brand new viewpoint on workforce activities on R-trees. They build a bunch RF(G), outfitted with an motion on an R-tree, whose components are definite features from a compact genuine period to the gang G. additionally they examine the constitution of RF(G), together with a close description of centralizers of components and an research of its subgroups and quotients. Any team appearing freely on an R-tree embeds in RF(G) for a few collection of G. a lot is still performed to appreciate RF(G), and the huge checklist of open difficulties integrated in an appendix might probably result in new tools for investigating crew activities on R-trees, fairly loose activities. This e-book will curiosity all geometric crew theorists and version theorists whose examine includes R-trees.

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Suppose that ( f ◦ g) ◦ h is defined, that is, ε0 ( f , g) = ε0 ( f ◦ g, h) = 0. 9, ε0 (g, h) = 0. 17(ii) and the remark then gives ε0 ( f , g ◦ h) = ε0 ( f , gh) = 0. Hence, f ◦ (g ◦ h) is defined. In a similar way one finds that ( f ◦ g) ◦ h is defined once f ◦ (g ◦ h) is defined. Finally, if both products are defined then they are equal by the associativity of the star operation. It remains to consider the case where L(g) = 0. Again suppose that ( f ◦ g) ◦ h is defined, implying ε0 ( f ◦ g, h) = 0.

The particular statement follows from this together with part (i). Example and Remark It is easy to see that reduced multiplication is not associative on the whole of F (G). For instance, let f , g be the functions, of lengths 1 and 12 , respectively, given by ⎧ x, ⎪ ⎪ ⎪ ⎨ f (ξ ) := 1G , ⎪ ⎪ ⎪ ⎩ −1 x , ⎫ 0 ≤ ξ < 12 ⎪ ⎪ ⎪ ⎬ 1 ξ=2 ⎪ ⎪ ⎪ ⎭ 1 < ξ ≤ 1 2 and g(ξ ) := ⎧ ⎨x, ⎫ 0 ≤ ξ < 12 ⎬ ⎩1 , G ξ= 1 2 ⎭ , where x is some fixed element of G. Certainly f = 1G . Moreover, for every ε in the range 0 ≤ ε ≤ 12 , we have f (1 − δ )g(δ ) = 1G , 0 ≤ δ ≤ ε; hence E ( f , g) = 0, 12 The group RF (G) 22 and ε0 ( f , g) = sup E ( f , g) = 12 .

27 The only bounded subnormal subgroup of RF (G) is the trivial group {1G }. Proof Let N ≤ RF (G) be a non-trivial, bounded, and subnormal subgroup of RF (G). Then G = {1G }, so G0 < RF (G) (for instance, RF (G) contains functions of positive length with constant value equal to some element x ∈ G − {1G }). 25 we have N ≤ tG0t −1 for some t ∈ RF (G). Let N = N0 N1 · · · Nr RF (G) be a strict subnormal series connecting N with RF (G). 4 Conjugating by t, we find that N t ≤ G0 and 4 Consider, for instance, a function f of length 1 given by f (ξ ) = x, 0≤ξ <1 1G , ξ = 1 (0 ≤ ξ ≤ 1), where x is some non-trivial element of G.

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A Universal Construction for Groups Acting Freely on Real Trees by Ian Chiswell

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