Current twisting and   nonsingular matrices

Clay, Matt; Pettet, Alexandra

doi:10.4171/cmh/257

Cited by 14 publications

(23 citation statements)

References 27 publications

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“…The proposition follows from a slight modification of the arguments from Theorem 5.2 in [12]. In its proof (equation (5.9)), we showed that for for every ǫ > 0 and integer r > 0, there is an N > 0 such that for n ≥ N:…”

Section: Dehn Twistsmentioning

confidence: 96%

“…These types of outer automorphisms were considered in [13] and used in [12] to show that there is no homological obstruction to full irreducibility. We briefly recall the setup here; for more details consult either of the references [12,13].…”

Section: Examplementioning

confidence: 99%

“…For large enough n, the elements φ n are fully irreducible and hyperbolic [ . The continuity of the Kapovich-Lustig intersection form (see [32] for its definition and properties) implies that c 1 has a fixed point in an accumulation point of the sequence {T + (ψ n )} (see [12,Remark 5.3]). Therefore as c 1 and c 2 are not simultaneously elliptic, c 2 has positive translation length in any such accumulation point.…”

Section: Dehn Twistsmentioning

confidence: 99%

“…Let T T 1 and T T 2 be the Cayley trees for the basis T 1 and T 2 , respectively. See [12,13] for the details underlying these constructions. Proposition 6.3.…”

Section: Dehn Twistsmentioning

confidence: 99%

“…Each of Sections 3 and 4 conclude with a proposition essential to the proofs of the main theorems, found in Section 5. In Section 6, we bring together results from Section 5 and previous papers of the authors [12,13] to describe a method for constructing geodesic axes of fully irreducible elements which enter the thin part of Outer space.…”

Section: Introductionmentioning

confidence: 99%

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Relative Twisting in Outer Space

Clay

Pettet

2012

J. Topol. Anal.

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Abstract. Subsurface projection is indispensable to studying the geometry of the mapping class group and the curve complex of a surface. When the subsurface is an annulus, this projection is sometimes called relative twisting. We give two alternate versions of relative twisting for the outer automorphism group of a free group. We use this to describe sufficient conditions for when a folding path enters the thin part of Culler-Vogtmann's Outer space. As an application of our condition, we produce a sequence of fully irreducible outer automorphisms whose axes in Outer space travel through graphs with arbitrarily short cycles; we also describe the asymptotic behavior of their translation lengths.

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Section: Dehn Twistsmentioning

confidence: 96%

Section: Examplementioning

confidence: 99%

Section: Dehn Twistsmentioning

confidence: 99%

“…Let T T 1 and T T 2 be the Cayley trees for the basis T 1 and T 2 , respectively. See [12,13] for the details underlying these constructions. Proposition 6.3.…”

Section: Dehn Twistsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Relative Twisting in Outer Space

Clay

Pettet

2012

J. Topol. Anal.

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Subset currents on free groups

Kapovich

Nagnibeda

2012

Geom Dedicata

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We introduce and study the space of \emph{subset currents} on the free group $F_N$. A subset current on $F_N$ is a positive $F_N$-invariant locally finite Borel measure on the space $\mathfrak C_N$ of all closed subsets of $\partial F_N$ consisting of at least two points. While ordinary geodesic currents generalize conjugacy classes of nontrivial group elements, a subset current is a measure-theoretic generalization of the conjugacy class of a nontrivial finitely generated subgroup in $F_N$, and, more generally, in a word-hyperbolic group. The concept of a subset current is related to the notion of an "invariant random subgroup" with respect to some conjugacy-invariant probability measure on the space of closed subgroups of a topological group. If we fix a free basis $A$ of $F_N$, a subset current may also be viewed as an $F_N$-invariant measure on a "branching" analog of the geodesic flow space for $F_N$, whose elements are infinite subtrees (rather than just geodesic lines) of the Cayley graph of $F_N$ with respect to $A$.Comment: updated version; to appear in Geometriae Dedicat

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Generalized north–south dynamics on the space of geodesic currents

Uyanik

2014

Geom Dedicata

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We prove uniform north-south dynamics type results for the action of ϕ ∈ Out(F N ) on the space of projectivized geodesic currents PCurr(S) = PCurr(F N ), where ϕ is induced by a pseudo-Anosov homeomorphism on a compact surface S with boundary such that π 1 (S) = F N . As an application, we show that for a subgroup H ≤ Out(F N ), containing a fully irreducible (iwip) element, either H contains a hyperbolic iwip or H is contained in the image in Out(F N ) of the mapping class group of a surface with a single boundary component.

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Current twisting and nonsingular matrices

Abstract: Abstract. We show that for k ≥ 3, given any matrix in GL(k, Z), there is a hyperbolic fully irreducible automorphism of the free group of rank k whose induced action on Z k is the given matrix.

Cited by 14 publications

References 27 publications

Relative Twisting in Outer Space

Relative Twisting in Outer Space

Subset currents on free groups

Generalized north–south dynamics on the space of geodesic currents

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