By Ian Chiswell
The speculation of R-trees is a well-established and demanding quarter of geometric crew conception and during this publication the authors introduce a building that gives a brand new viewpoint on team activities on R-trees. They build a gaggle RF(G), outfitted with an motion on an R-tree, whose components are yes features from a compact genuine period to the gang G. additionally 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 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 might almost certainly result in new tools for investigating team activities on R-trees, quite unfastened activities. This booklet will curiosity all geometric workforce theorists and version theorists whose learn comprises R-trees.
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Additional resources for A Universal Construction for Groups Acting Freely on Real Trees
15, using the geometry of R-trees, as follows. (ii) Suppose that f k = gk is hyperbolic. 54(i), we have ( f ) = (g) > 0; thus f , g are hyperbolic and f , g have the same axis. 14. 54(i). Consequently, [r, s] ﬁxes every point of this common axis; hence, as before, [r, s] = 1G , as claimed. 20 If f ∈ RF (G) then we have x0 ∈ A f if and only if f is cyclically reduced. 28). 1, we have cx0 ( f , g) = ε0 ( f −1 , g). Hence we conclude that x0 ∈ A f if and only if ε0 ( f , f ) = 0; that is, if and only if f is cyclically reduced.
Then f −1 = f2−1 ∗ f1−1 ; in particular, f = f1 ◦ f2 implies f −1 = f2−1 ◦ f1−1 . Proof (i) We note that f 1 (L( f1 )) f 2 (0) = f2−1 (L( f2−1 )) f1−1 (0) −1 ; in particular, f1 (L( f1 )) f 2 (0) = 1G ⇐⇒ f2−1 (L( f2−1 )) f1−1 (0) = 1G . Thus, if f1 (L( f1 )) f2 (0) = 1G then ε0 ( f1 , f2 ) = ε0 ( f2−1 , f1−1 ) = 0; hence, we may assume that f1 (L( f1 )) f2 (0) = 1G , implying ε0 ( f1 , f2 ) = sup E ( f1 , f2 ) as well as ε0 ( f2−1 , f1−1 ) = sup E ( f2−1 , f1−1 ). 2 Reduced functions and reduced multiplication that is, E ( f1 , f2 ) = E ( f2−1 , f1−1 ) and thus ε0 ( f1 , f2 ) = ε0 ( f2−1 , f1−1 ), as claimed.
7) we conclude that ε0 + ε ∈ E ( f , g), implying ε ≤ 0, a contradiction. Hence, f g is reduced as claimed. As a consequence of the product deﬁnition plus the fact that the product of two reduced functions is also reduced, we have the following. 8 For f , g ∈ RF (G), the following assertions are equivalent: (i) ε0 ( f , g) = 0; (ii) f g = f ∗ g; (iii) f ∗ g is reduced. Proof (i) ⇒ (ii). 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.
A Universal Construction for Groups Acting Freely on Real Trees by Ian Chiswell