By Ian Chiswell

The idea of R-trees is a well-established and demanding zone of geometric staff idea and during this ebook the authors introduce a building that offers a brand new point of view on team activities on R-trees. They build a bunch RF(G), outfitted with an motion on an R-tree, whose components are yes capabilities from a compact genuine period to the crowd G. in addition they research the constitution of RF(G), together with a close description of centralizers of parts and an research of its subgroups and quotients. Any team appearing freely on an R-tree embeds in RF(G) for a few number of G. a lot is still performed to appreciate RF(G), and the huge checklist of open difficulties incorporated in an appendix might possibly result in new equipment for investigating crew activities on R-trees, relatively loose activities. This booklet will curiosity all geometric crew theorists and version theorists whose examine contains R-trees.

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2. Give the detailed working to show that, for f ∈ F (G), we have ε0 ( f , f −1 ) = L( f ) = ε0 ( f −1 , f ) and that, consequently, f f −1 = 1G = f −1 f . 3. 17(ii) in full. 4. Let G be a group, and suppose that G ∼ = G1 ∗ G2 , that is, G is the free product of two subgroups G1 and G2 . Show that the subgroup H := RF (G1 ), RF (G2 ) of RF (G) satisﬁes H ∼ = H1 ∗ H2 where, for i = 1, 2, Hi := RF (Gi ). 1 Introduction In this chapter we explore the geometry associated with RF (G) via its length function L.

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. 25. If f ∈ RF (G) is a nontrivial torsion element then f , the cyclic subgroup generated by f , is ﬁnite and thus conjugate to a subgroup of G0 .

7 (i) Let f ∈ RF (G). Then there exist t, f 1 ∈ RF (G), with f1 cyclically reduced, such that f = t ◦ f 1 ◦ t −1 . 6) where t, s, f 1 , f2 ∈ RF (G) and f1 , f2 are cyclically reduced, then s = tg and f2 = g−1 f 1 g for some g ∈ G0 ; in particular, L( f1 ) = L( f2 ). Proof (i) For f ∈ RF (G), set ε0 ( f ) := min ε0 ( f , f ), 12 L( f ) . 14 (dissection of reduced functions), we can ﬁnd t, f0 ∈ RF (G) such that f = t ◦ f0 and L(t) = ε0 ( f ). 14 again to obtain f0 = f1 ◦ u where L(u) = ε0 ( f ). 18 (associativity of the circle product) we have ε0 (t, f1 ) = 0, so that t ◦ f1 is deﬁned and L(t ◦ f1 ) = L(t) + L( f1 ) = L( f ) − ε0 ( f ).