Download Combinatorial and Geometric Group Theory: Dortmund and by Oleg Bogopolski, Inna Bumagin, Olga Kharlampovich, Enric PDF

By Oleg Bogopolski, Inna Bumagin, Olga Kharlampovich, Enric Ventura

This quantity assembles a number of examine papers in all parts of geometric and combinatorial staff thought originated within the contemporary meetings in Dortmund and Ottawa in 2007. It comprises top of the range refereed articles developping new points of those sleek and lively fields in arithmetic. it's also acceptable to complicated scholars drawn to fresh effects at a examine point.

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1. We begin by marking all edges in the initial slice ρ0 and in all notches μi , νi with their height. Assume inductively that all edges in a slice ρi−1 have been marked, and let E be an edge of height r in ρi−1 , with marking s. Now, consider an edge E in f (E). If the height of E is r, or if Hs is a zero stratum, then we keep the marking s. If the height of E is less than r, then we mark E by r. This gives us a marking for all edges in μi f (ρi−1 )νi . 1) may give rise to different possible markings, but this will not be a problem.

Intuitively, this then f# means that if few legal segments disappear, then many Nielsen paths will appear. Since N (ρ) only counts those legal segments that do not overlap with a Nielsen path, this observation will yield the desired estimate. , we have N (γ) = n(γ). If N (γ) ≥ 4, then for every four consecutive M M legal segments whose images do not cancel completely in f# (γ), f# (γ) contains 6 M at least one Nielsen subpath, so that we have N (f# (γ)) ≤ 7 N (γ), using the same reasoning as above.

Let μ1 , μ2 be Nielsen paths in G, and let ν be some path in G. • If μ1 and μ2 can be concatenated, then the path obtained from μ1 μ2 by tightening relative endpoints is also a Nielsen path. 38 P. Brinkmann • If μ1 and ν can be concatenated, let γ be the path obtained by tightening μ1 ν, and let Δ = L(γ) − L(ν). Then, for all k ≥ 0, we have k k (γ) = L f# (ν) + Δ L f# and − L(μ1 ) ≤ Δ ≤ L(μ1 ). 9 for automorphisms of linear growth. 9. 2. Assume that f : G → G is of linear growth. 9 holds with K = 1.

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