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By Ian Chiswell

The idea of R-trees is a well-established and demanding zone of geometric staff thought and during this booklet the authors introduce a building that offers a brand new point of view on team activities on R-trees. They build a gaggle RF(G), outfitted with an motion on an R-tree, whose components are sure services from a compact actual period to the gang G. additionally they research 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 wide record of open difficulties incorporated in an appendix may probably bring about new tools for investigating staff activities on R-trees, quite loose activities. This publication will curiosity all geometric crew theorists and version theorists whose examine comprises R-trees.

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Extra resources for A Universal Construction for Groups Acting Freely on Real Trees

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22 Let G = {1, ζ } be a cyclic group of order 2. Define an equivalence relation on subsets of the set of natural numbers N via A ∼ B : ⇐⇒ |AΔB| < ∞, 34 The group RF (G) and choose a system of representatives for the equivalence classes such that finite sets are represented by the empty set. Then we define the product ∏∞ ν=0 gν of a sequence (gν )ν≥0 in G as follows: let A be the set of indices ν with gν = ζ , |AΔB| . This infinite prodlet B be the representative of A, and set ∏∞ ν=0 gν = ζ uct concept satisfies the usual requirements, for instance the empty product equals 1 and changing finitely many terms changes the product in the expected way, but it is clearly highly unnatural since it depends heavily on the choice of representatives.

19. 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 .

15, L( f n ) = 2L(t) + nL( f1 ) → ∞ as n → ∞, contradicting assertion (i). Finally, suppose that H does not contain a hyperbolic element. In showing that (ii) implies (iii), we may suppose that H = {1G }. Fix a non-trivial element a = t ◦ g ◦ t −1 in H with L(g) = 0, and let b ∈ H be an arbitrary element. 8, a lies in tG0t −1 and in no other conjugate of G0 . 22 implies that b ∈ tG0 t −1 ; so H ⊆ tG0t −1 , since b is arbitrary. 26 Every finite subgroup of RF (G) is conjugate to a subgroup of G0 ; in particular, RF (G) is torsion-free if and only if G is torsion-free.

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