By Ian Chiswell
The idea of R-trees is a well-established and significant quarter of geometric crew conception and during this booklet the authors introduce a development that offers a brand new standpoint on workforce 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 actual period to the gang 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 crew 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 huge record of open difficulties integrated in an appendix may possibly possibly result in new equipment for investigating team activities on R-trees, fairly loose activities. This booklet will curiosity all geometric crew theorists and version theorists whose learn comprises R-trees.
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Additional info for A Universal Construction for Groups Acting Freely on Real Trees
Then there exist reduced functions f1 : [0, β ] → G and f2 : [0, α − β ] → G such that f = f1 ◦ f2 . Moreover, f1 and f2 with these properties are uniquely determined once one of the values f1 (β ), f2 (0) has been speciﬁed; one of these values may be chosen arbitrarily in G. 3 Cancellation theory for RF (G) 23 Proof The equation f = f1 ∗ f2 with L( f1 ) = β and L( f2 ) = α − β is equivalent to the conjunction of f1 (ξ ) = f (ξ ), 0 ≤ ξ < β, f2 (ξ ) = f (ξ + β ), 0 < ξ ≤ α −β, and f1 (β ) f2 (0) = f (β ).
Case 2(b): L(u1 ) = 0. Changing the values g2 (L(u)), v(0), and h1 (0) if necessary, we can ensure that g2 = u−1 while keeping the equations g = g2 ◦ v and h = v−1 ◦ h1 intact, the former now reading g = u−1 ◦ v. 1 that v = g1 in this case. 16, ( f g)h = ( f1 ◦ g1 )(v−1 ◦ h1 ) = ( f1 ◦ v)(v−1 ◦ h1 ) = f 1 h1 and f (gh) = ( f1 ◦ u)(g2 ◦ h1 ) = ( f1 ◦ u)(u−1 ◦ h1 ) = f 1 h1 , completing the proof of the theorem. 19 Given a group G, we set G0 := f ∈ RF (G) : L( f ) = 0 , a subgroup of RF (G) canonically isomorphic to G.
7 we have s = tk for some k ∈ G0 , so sG0 s−1 = tG0t −1 . 8 may be rephrased by saying that the set tG0t −1 : t ∈ RF (G) of conjugates of G0 forms an amalgam with trivial intersection. 7 for given f ∈ RF (G), which is unique up to conjugation by a G0 -element, is called the (cyclically reduced) core of f , denoted c( f ). The passage from f to f 1 itself is termed cyclic reduction of f . (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 .