By Ian Chiswell

ISBN-10: 1107024811

ISBN-13: 9781107024816

The speculation of R-trees is a well-established and significant region of geometric workforce conception and during this ebook the authors introduce a building that gives a brand new point of view on staff activities on R-trees. They build a gaggle RF(G), built with an motion on an R-tree, whose components are definite services from a compact genuine period to the crowd 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 staff 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 broad checklist of open difficulties incorporated in an appendix may possibly probably bring about new equipment for investigating team activities on R-trees, relatively unfastened activities. This publication will curiosity all geometric staff theorists and version theorists whose learn comprises R-trees.

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

**Example text**

3) implies that f k is cyclically reduced for all k ∈ Z, since L( f 2k ) = 2|k|L( f ) = 2L( f k ). 3). Let f ∈ RF (G) be cyclically reduced. Then we claim that fk = f ◦ f ◦···◦ f, k ∈ N0 . 4) k times This is trivial if L( f ) = 0; thus, we may suppose that L( f ) > 0. 4) holds trivially for k = 0, 1, 2. 4) holds with k replaced by k − 1 and that k ≥ 3. If ε0 ( f k−1 , f ) were strictly positive then there would exist ε such that 0 < ε < L( f ) and f k−1 (L( f k−1 ) − η) f (η) = 1G , 0 ≤ η ≤ ε. 4 Cyclic reduction 41 which contradicts our assumption that f is cyclically reduced.

If ε0 ( f , g) = 0 then L( f g) = L( f ) + L(g) = L( f ∗ g), and, by the deﬁnitions of f g and ⎧ f (ξ ), ⎪ ⎪ ⎪ ⎨ ( f g)(ξ ) = f (L( f ))g(0), ⎪ ⎪ ⎪ ⎩ g(ξ − L( f )), f ∗ g, we have, for 0 ≤ ξ ≤ L( f ) + L(g), that ⎫ 0 ≤ ξ < L( f ) ⎪ ⎪ ⎪ ⎬ = ( f ∗ g)(ξ ); ξ = L( f ) ⎪ ⎪ ⎪ ⎭ L( f ) < ξ ≤ L( f ) + L(g) hence, f g = f ∗ g as claimed. (ii) ⇒ (iii). 7. (iii) ⇒ (i). Suppose that ε0 ( f , g) > 0. Then α := L( f ) and β := L(g) are strictly positive, f (α) = g(0)−1 , and ε0 ( f , g) = sup E ( f , g); in particular, α is an interior point of the interval [0, α + β ] and ( f ∗ g)(α) = 1G .

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 . Hence G itself contains a non-trivial torsion element. Conversely, if G contains a non-trivial torsion element then so does G0 , and hence RF (G).

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

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