By Ian Chiswell
The speculation of R-trees is a well-established and significant zone of geometric workforce conception and during this publication the authors introduce a building that offers a brand new viewpoint on crew activities on R-trees. They build a bunch RF(G), outfitted with an motion on an R-tree, whose parts are sure capabilities from a compact genuine period to the crowd G. in addition they examine the constitution of RF(G), together with a close description of centralizers of parts and an research of its subgroups and quotients. Any staff appearing freely on an R-tree embeds in RF(G) for a few collection of G. a lot continues to be performed to appreciate RF(G), and the huge checklist of open difficulties integrated in an appendix may perhaps in all probability result in new equipment for investigating workforce activities on R-trees, fairly loose activities. This e-book will curiosity all geometric crew theorists and version theorists whose examine includes R-trees.
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Extra info for A Universal Construction for Groups Acting Freely on Real Trees
Hence, f must be hyperbolic. 8 can now be rephrased as follows: every nontrivial elliptic element of RF (G) lies in precisely one conjugate of G0 . Equivalently, every non-trivial elliptic element stabilises exactly one point of XG . 20. If [x, y] is a segment in a Λ-tree X = (X, d) with x = y and G is a group acting on X by isometries, then the pointwise stabiliser of [x, y] is stab(x) ∩ stab(y); such a subgroup of G will be called a pointwise arc stabiliser. 14, every (pointwise) arc stabiliser is trivial.
Hence f1 and f2 living on the domains speciﬁed above and solving the equation f = f1 ∗ f2 do exist; they are uniquely determined once one of the values f1 (β ), f2 (0) has been speciﬁed, and one of these values can be chosen arbitrarily. 8. 15 (Visibility of cancellation) Let f , g ∈ RF (G) be reduced functions. Then there exist f1 , g1 , u ∈ RF (G) such that f = f1 ◦u, g = u−1 ◦g1 , and f g = f1 ◦ g1 . Proof If ε0 := ε0 ( f , g) = 0 then our conclusion is satisﬁed for f1 := f , g1 := g, and u := 1G ; hence, we may assume that ε0 > 0.
Moreover we are at liberty to deﬁne w(L(u)) as we like. Making use of the equations g2 = w ◦ g3 , g = g2 ◦ v, and g = u−1 ◦ g1 , we ﬁnd that w(ξ ) = g2 (ξ ) = g(ξ ) = u−1 (ξ ), 0 ≤ ξ < L(u), and, deﬁning w(L(u)) := u−1 (L(u)), we obtain w = u−1 , as required. Note that L(g3 ) = L(g2 ) − L(u) > 0 by the case assumption. 18 (associativity of the circle product) in the last step. 1. 18 now yields f g = f1 ◦ g1 = f1 ◦ (g3 ◦ v) = ( f1 ◦ g3 ) ◦ v; in particular, ε0 ( f1 , g3 ) = 0. 16 (visible cancellation) gives ( f g)h = (( f1 ◦ g3 ) ◦ v)(v−1 ◦ h1 ) = ( f1 ◦ g3 )h1 .
A Universal Construction for Groups Acting Freely on Real Trees by Ian Chiswell