By Ian Chiswell

ISBN-10: 1107024811

ISBN-13: 9781107024816

The idea of R-trees is a well-established and critical zone of geometric workforce idea and during this booklet the authors introduce a development that offers a brand new standpoint on workforce activities on R-trees. They build a gaggle RF(G), outfitted with an motion on an R-tree, whose parts are definite capabilities from a compact genuine period to the gang G. in addition they learn 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 number of G. a lot is still performed to appreciate RF(G), and the broad checklist of open difficulties integrated in an appendix may in all probability bring about new tools for investigating team activities on R-trees, rather loose activities. This booklet will curiosity all geometric crew theorists and version theorists whose learn consists of R-trees.

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Then the map ϕ deﬁnes a notion of an inﬁnite product in G in the following way. Given a sequence of elements {gν }ν≥0 in G and a ﬁxed element g ∈ G, deﬁne a function f ∈ F (G) of length 1 via ⎧ ⎪ gν , ξ = 1 − 2−ν , ν ≥ 0, ⎪ ⎪ ⎪ ⎪ ⎪ 1 − 2−ν < ξ < 1 − 3 × 2−ν−2 , ν ≥ 0, ⎪ ⎨g, f (ξ ) = 1G , ξ = 1 − 3 × 2−ν−2 , ν ≥ 0, ⎪ ⎪ ⎪ ⎪g−1 , 1 − 3 × 2−ν−2 < ξ < 1 − 2−ν−1 , ν ≥ 0, ⎪ ⎪ ⎪ ⎩1 , ξ = 1. G Then L(ϕ( f )) = 0 and we can set ∞ ∏ gν := ϕ( f )(0). ν=0 However, in general G will not admit a natural and useful concept of inﬁnite product (compare, for instance, the next example); hence, in general a reasonable reduction process ϕ does not exist.

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).

G Then L(ϕ( f )) = 0 and we can set ∞ ∏ gν := ϕ( f )(0). ν=0 However, in general G will not admit a natural and useful concept of inﬁnite product (compare, for instance, the next example); hence, in general a reasonable reduction process ϕ does not exist. 22 Let G = {1, ζ } be a cyclic group of order 2. Deﬁne an equivalence relation on subsets of the set of natural numbers N via A ∼ B : ⇐⇒ |AΔB| < ∞, 34 The group RF (G) and choose a system of representatives for the equivalence classes such that ﬁnite sets are represented by the empty set.

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

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