By Ian Chiswell

The idea of R-trees is a well-established and significant quarter of geometric crew concept and during this e-book the authors introduce a development that gives a brand new viewpoint on staff activities on R-trees. They build a gaggle RF(G), built with an motion on an R-tree, whose parts are sure capabilities from a compact genuine period to the crowd G. additionally they learn the constitution of RF(G), together with a close description of centralizers of parts and an research of its subgroups and quotients. Any workforce appearing freely on an R-tree embeds in RF(G) for a few selection of G. a lot is still performed to appreciate RF(G), and the vast checklist of open difficulties incorporated in an appendix may well in all probability bring about new equipment for investigating staff activities on R-trees, relatively unfastened activities. This ebook will curiosity all geometric workforce theorists and version theorists whose examine consists of R-trees.

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

The idea of R-trees is a well-established and significant sector of geometric crew idea and during this ebook the authors introduce a building that gives a brand new point of view on workforce activities on R-trees. They build a gaggle RF(G), outfitted with an motion on an R-tree, whose components are sure capabilities from a compact actual period to the gang G.

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**Example text**

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 .

1 1 This equation is to be interpreted in the manner standard for partial operations: if one of the products ( f ◦ g) ◦ h and f ◦ (g ◦ h) is deﬁned then so is the other, and the two products are equal. 28 The group RF (G) Proof We ﬁrst consider the case where L(g) > 0. 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.

It also follows that the trivial group {1G } is the only bounded subnormal subgroup of RF (G). 2 Construction of XG The concept of a Lyndon length function. 3. From now on, the term length function will always mean Lyndon length function. We are concerned with real-valued length functions L : Γ → R, where Γ is a group, and as usual we deﬁne c(γ1 , γ2 ) = 1 2 L(γ1 ) + L(γ2 ) − L(γ1−1 γ2 ) for γ1 , γ2 ∈ Γ. 1 The map L : RF (G) → R which assigns to each function f ∈ RF (G) the length L( f ) of its domain, that is, the interval [0, L( f )], is a (real) length function.