Relative Twisting in Outer Space

Clay, Matt; Pettet, Alexandra

doi:10.1142/s1793525312500100

Cited by 9 publications

(10 citation statements)

References 31 publications

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“…The arguments given below prove also that L 2 BF H is contained in supp(µ − ). Indeed, a direct argument shows that the two laminations are equal, see [6] or [7].…”

Section: Introductionmentioning

confidence: 99%

Invariant Laminations for Irreducible Automorphisms of Free Groups

Kapovich

Lustig

2014

The Quarterly Journal of Mathematics

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For every irreducible hyperbolic automorphism ϕ of FN (i.e. the analogue of a pseudo-Anosov mapping class) it is shown that the algebraic lamination dual to the forward limit tree T+(ϕ) is obtained as "diagonal closure" of the support of the backward limit current µ−(ϕ). This diagonal closure is obtained through a finite procedure in analogy to adding diagonal leaves from the complementary components to the stable lamination of a pseudo-Anosov homeomorphism. We also give several new characterizations as well as a structure theorem for the dual lamination of T+(ϕ), in terms of Bestvina-Feighn-Handel's "stable lamination" associated to ϕ.

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“…The arguments given below prove also that L 2 BF H is contained in supp(µ − ). Indeed, a direct argument shows that the two laminations are equal, see [6] or [7].…”

Section: Introductionmentioning

confidence: 99%

Invariant Laminations for Irreducible Automorphisms of Free Groups

Kapovich

Lustig

2014

The Quarterly Journal of Mathematics

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“…Remark The twisting number defined above is a rational number. The integer part of

{prefixtwist}_{α} false(T, T_{0} false)

, which we denote by

{prefixtw}_{α} false(T, T_{0} false)

, is equal to the Clay–Pettet definition of relative twisting number which considers hyperplanes

τ, τ_{0}

in the core that are dual to specific edges and counts how many

α

–translates of

τ

intersects

τ_{0}

(see [8] for details). Similar to the above example, one can see that in the example of Theorem A,

{prefixtw}_{b} false(R_{0}, ϕ (R_{0}) false) = s - 1 and {prefixtw}_{a b^{s}} false(R_{0}, ϕ (R_{0}) false) = t - 1 .

…”

Section: Twisting Estimatementioning

confidence: 99%

“…On the other hand, we observe in the above example, that even along the shorter paths, it takes at least

s

steps to form or to eliminate an

s

th power of

b

and

t

steps to eliminate a

t

th power of

(a b^{s})

. We examine this phenomenon through the language of relative twisting number [8].…”

Section: Introductionmentioning

confidence: 99%

“…The relative twisting number [8] of two labelled graphs

x

and

x^{'}

around a loop

α

measures the difference between

x

and

x^{'}

from the point of view of

α

and is denoted by

{prefixtw}_{α} false(x, x^{'} false)

(see Definition 3.1). We show that the length of any path in

Out ({double-struckF}_{n})

connecting

ϕ

to the identity is bounded below by the relative twisting number

{prefixtw}_{α} false(R, R_{0} false)

, where

R = ϕ (R_{0})

.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quasi‐geodesics in Out(Fn) and their shadows in sub‐factors

Qing

Rafi

2021

J. London Math. Soc.

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We study the behaviour of quasi-geodesics in Out (Fn). Given an element φ in Out(Fn), there are several natural paths connecting the origin to φ in Out(Fn); for example, paths given by Stallings' folding algorithm and paths induced by the shadow of greedy folding paths in Outer Space. We show that none of these paths is, in general, a quasi-geodesic in Out(Fn). In fact, in contrast with the mapping class group setting, we construct examples where any quasi-geodesic in Out(Fn) connecting φ to the origin will have to backtrack in some free factor of Fn.

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“…Still, Guirardel's intersection number is a highly useful tool when studying the asymptotic geometry of cv N itself, particularly when looking at orbits of subgroups of Out(F N ) in cv 1 N and cv N . Examples of such applications can be found in [4,12,13,14,29,32].…”

Section: Introductionmentioning

confidence: 99%