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

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 includes top of the range refereed articles developping new elements of those sleek and energetic fields in arithmetic. it's also acceptable to complicated scholars attracted to fresh effects at a examine point.

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If Hr supports a closed Nielsen path τ , then the initial and terminal edges of τ are partial edges in Hr , and we may assume that the image of each of them also contains at least L edges in Hr . We say that a legal path of height r is long if it contains at least L edges in Hr . 4. 1. Let Hr be an exponentially growing stratum or a fast polynomial stratum. Then there exists a computable constant λ > 1 such that if σ is a circuit in Gr or a path starting and ending at fixed vertices, then either σ is a concatenation of Nielsen paths of height r and subpaths in Gr−1 , or we have k (σ)) ≥ λk L(σ) + L(f# for all k ≥ 0.

Brinkmann determined, but the sequence is not. For instance, EE −1 E may be tightened as E(E −1 E) or (EE −1 )E. 2. Let ρi , i = 1, . . , k be paths that can be concatenated to form a path ρ = ρ1 ρ2 · · · ρk . When tightening f (ρ) to obtain f# (ρ), we adopt the convention that we first tighten the images of ρi to f# (ρi ). In a second step, we tighten the concatenation f# (ρ1 ) · · · f# (ρk ) to f# (ρ). In many situations, the length of a subpath ρi will be greater than the number of edges that cancel at either end, in which case it makes sense to talk about edges in f# (ρ) originating from ρi .

The first four properties follow immediately from definitions. In order to prove the fifth property, we just remark that each element of M2 has at most 2CD(ρ) visible edges that do not appear in ρ itself. Since M2 contains at most 2CD(ρ) hallways, the estimate follows. 4. There exists a (computable) constant C with the following property: Let γ be a path of height r, starting and ending at vertices, and assume that Er is of degree d > 1. Then, for all k ≥ 0, k L(γ) + L(f# (γ)) ≥ Ck d . Proof. It suffices to prove the lemma if either γ = Er γ , or γ = Er γ Es−1 , where γ only involves edges of degree less than d, and Es is of degree d.

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