Download PDF by Bogopolski O., et al. (eds.): Combinatorial and geometric group theory: Dortmund and

By Bogopolski O., et al. (eds.)

ISBN-10: 3764399104

ISBN-13: 9783764399108

This quantity assembles numerous learn papers in all parts of geometric and combinatorial team concept originated within the contemporary meetings in Dortmund and Ottawa in 2007. It includes prime quality refereed articles developping new features of those sleek and lively fields in arithmetic. it's also applicable to complicated scholars drawn to fresh effects at a examine point.

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Example text

The result of such a sequence of cancellations is uniquely 22 P. 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 .

In many cases, it is convenient to work with outer automorphisms. Topologically, this means that we work with homotopy equivalences rather that based homotopy equivalences. , there exist subgraphs G0 = ∅ ⊂ G1 ⊂ · · · ⊂ Gk = G such that for each filtration element Gr , the restriction of f to Gr is a homotopy equivalence of Gr . The subgraph Hr = Gr \ Gr−1 is called the rth stratum of the filtration. We say that a path ρ has nontrivial intersection with a stratum Hr if ρ crosses at least one edge in Hr .

The stratum {x, y} grows exponentially, and we have ψ(xyx−1 y −1 ) = xyx−1 y −1 . This means that a grows linearly although it maps across an exponentially growing stratum. This is another phenomenon that we need to consider when analyzing strata of linear growth. The notion of hallways naturally extends to mapping tori of homotopy equivalences of finite graphs. Specifically, a hallway ρ in the mapping torus of f : G → G is a sequence of paths of the form ρ = (μk−1 , μk−2 , . . , μ1 , ρ0 , ν1 , ν2 , .

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Combinatorial and geometric group theory: Dortmund and Ottawa-Montreal conf. by Bogopolski O., et al. (eds.)

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