By Ian Chiswell
The idea of R-trees is a well-established and critical sector of geometric team concept and during this ebook the authors introduce a building that offers a brand new point of view on staff activities on R-trees. They build a bunch RF(G), built with an motion on an R-tree, whose parts are definite features 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 performing 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 large checklist of open difficulties incorporated in an appendix may perhaps very likely result in new equipment for investigating staff activities on R-trees, quite unfastened activities. This publication will curiosity all geometric team 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
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 . Moreover, there exists ε ∈ E ( f , g) with ε > 0 and, for this ε, we have f (α − η)g(η) = 1G , 0 ≤ η ≤ ε. 8) can be rewritten as ( f ∗ g)(α − η)( f ∗ g)(α + η) = 1G , 0 < η ≤ ε. 9) says that [α − ε, α + ε] is a cancelling neighbourhood for f ∗ g around the interior point α, so that f ∗ g is not reduced, contradicting assertion (iii).
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. 7) then ( f ) = L( f1 ); that is, we have ( f ) = L(c( f )), f ∈ RF (G). 13. 18. 22 Let a and b be two elliptic elements of RF (G) which do not lie in the same conjugate of G0 .
Ii) The extra assumption that c( f )(0) = 1G singles out a uniquely deﬁned core c0 ( f ) of f ∈ RF (G), termed the normalized core of f . The function c0 ( f ) exists, and is uniquely determined by f through the conditions that f = t ◦ c0 ( f ) ◦ t −1 for some t ∈ RF (G), that c0 ( f ) is cyclically reduced, and that c0 ( f )(0) = 1G . 5 for more details. An element γ ∈ Γ is called • elliptic, if γ has a ﬁxed point, • an inversion, if γ has no ﬁxed point but γ 2 has a ﬁxed point, • hyperbolic, if γ is neither elliptic nor an inversion.
A Universal Construction for Groups Acting Freely on Real Trees by Ian Chiswell