By Ian Chiswell

ISBN-10: 1107024811

ISBN-13: 9781107024816

The speculation of R-trees is a well-established and critical region of geometric workforce thought and during this ebook the authors introduce a building that gives a brand new viewpoint on workforce activities on R-trees. They build a gaggle RF(G), built with an motion on an R-tree, whose parts are definite features from a compact actual period to the gang G. additionally they learn the constitution of RF(G), together with a close description of centralizers of components and an research of its subgroups and quotients. Any staff performing 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 vast checklist of open difficulties incorporated in an appendix may almost certainly result in new equipment for investigating workforce activities on R-trees, really loose activities. This booklet will curiosity all geometric team theorists and version theorists whose learn includes R-trees.

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**Extra info for A Universal Construction for Groups Acting Freely on Real Trees**

**Example text**

It follows that L( f g) = and that, for 0 ≤ ξ ≤ 12 , ⎧ ⎨ f (ξ ), 0 ≤ ξ < 12 , ( f g)(ξ ) = ⎩ f ( 1 )g( 1 ), ξ = 1 , 2 2 2 1 2 = ⎧ ⎨x, 0 ≤ ξ < 12 , ⎩1 , G ξ = 12 , = g(ξ ), that is, f g = g. We conclude that ( f g)g−1 = gg−1 = 1G = f = f (gg−1 ), which shows that reduced multiplication is indeed not associative on F (G). 3 Cancellation theory for RF (G) Our principal aim for the remainder of this chapter is to show that the restriction of reduced multiplication to the subset RF (G) is associative, so that we have the following.

7. 29 for more details. 1) 38 The R-tree XG associated with RF (G) and denote by f the equivalence class of f ∈ RF (G). One easily sees that f ≈ g ⇐⇒ L( f ) = L(g) and f |[0,L( f )) = g|[0,L(g)) ⇐⇒ f G0 = gG0 , so that RF (G)/ ≈ is nothing other than the coset space RF (G)/G0 . Next, we form the set YG := ( f , α) : f ∈ RF (G), α ∈ R, 0 ≤ α ≤ L( f ) , and introduce an equivalence relation ∼ on YG via ( f , α) ∼ ( g , β ) :⇐⇒ ε0 ( f −1 , g) ≥ α = β . We denote the equivalence class of ( f , α) by f , α , observing that we always have f , α = f |[0,α] , α .

FK+1 be functions such that L( f j ) > 0 for 2 ≤ j ≤ K, and set The group RF (G) 16 g := f1 ∗ · · · ∗ fK . Then, for 0 ≤ ξ ≤ ξK+1 , we have ( f1 ∗ · · · ∗ fK ∗ fK+1 )(ξ ) = (g ∗ fK+1 )(ξ ) = = = ⎧ g(ξ ), ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0 ≤ ξ < ξK , g(ξK ) fK+1 (0), ξ = ξK , fK+1 (ξ − ξK ), ξK < ξ ≤ ξK+1 , ⎧ ⎪ f1 (ξ ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ f j (ξ − ξ j−1 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ fK (ξ − ξK−1 ), 0 ≤ ξ < ξ1 , ξ j−1 < ξ < ξ j , 2 ≤ j ≤ K − 1, ξK−1 < ξ < ξK , ⎪ ⎪ ξ = ξ j , 1 ≤ j ≤ K − 1, f j (L( f j )) f j+1 (0), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ fK (L( fK )) fK+1 (0), ξ = ξK , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ξK < ξ ≤ ξK+1 , fK+1 (ξ − ξK ), ⎧ ⎪ f1 (ξ ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ f j (ξ − ξ j−1 ), 0 ≤ ξ < ξ1 , ξ j−1 < ξ < ξ j , 2 ≤ j ≤ K, ⎪ ⎪ fK+1 (ξ − ξK ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (L( f )) f (0), fj j j+1 ξK < ξ ≤ ξK+1 , ξ = ξ j , 1 ≤ j ≤ K.

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

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