# Get Combinatorial and geometric group theory: Dortmund and PDF

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

ISBN-10: 3764399104

ISBN-13: 9783764399108

This quantity assembles a number of examine papers in all parts of geometric and combinatorial staff concept originated within the contemporary meetings in Dortmund and Ottawa in 2007. It includes top of the range refereed articles developping new facets of those smooth and energetic fields in arithmetic. it's also applicable to complex scholars drawn to contemporary effects at a learn point.

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An unordered pair of edges in G originating from the same vertex is called a turn. A turn is called degenerate if the two edges are equal. We deﬁne a map Df : {turns in G} → {turns in G} by sending each edge in a turn to the ﬁrst edge in its image under f . A turn is called illegal if its image under some iterate of Df is degenerate, legal otherwise. An edge path ρ = E1 E2 · · · Es is said to contain the turns (Ei−1 , Ei+1 ) for 1 ≤ i < s. ρ is said to be legal if all its turns are legal, and a path ρ ⊂ Gr is r-legal if no illegal turn in α involves an edge in Hr .

Let f : G → G be a homotopy equivalence. There exists a constant Cf , depending only on f , with the property that for any tight path ρ in G obtained by concatenating two paths α, β, we have L(f# (ρ)) ≥ L(f# (α)) + L(f# (β)) − Cf . An upper bound for Cf can easily be read oﬀ from the map f [Coo87]. Let f : G → G be an improved relative train track map with an exponentially growing stratum Hr with growth rate λr . The r-length of a path ρ in G, Lr (ρ), is the total length of ρ ∩ Hr . 2)) will tend to inﬁnity as k tends to inﬁnity.

N (ρk+1 ) ≥ 4). If N (ρ1 ) < 4 and N (ρk+1 ) < 4, we have 6 N (ρ1 ) + N (ρk+1 ) + (N (ρ) − N (ρ1 ) − N (ρk+1 )) 7 6 6 ≤ 6 + (N (ρ) − 6) ≤ (1 + N (ρ)). 7 7 6 M Similar estimates yield that N (f# (ρ)) ≤ 7 (1 + N (ρ)) regardless of N (ρ1 ) and N (ρk+1 ). M If N (ρ) > 11, then 67 (1 + N (ρ)) ≤ 13 14 N (ρ), which implies that N (f# (ρ)) ≤ 13 N (ρ) if N (ρ) > 11, so that the ﬁrst inequality of the lemma holds with λ = 14 14 13 M and N0 = 11. As for the second inequality, we remark that N (f# (ρ)) ≤ N (ρ) and, if N (ρ) ≤ 11, then N (ρ) ≤ λ−1 N (ρ) + 1.