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(3)
Curve Complex
Free Group
Intersection Number
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Geometric intersection number and analogues of the curve complex for free groups
Geometric intersection number and analogues of the curve complex for free groups,10.2140/gt.2009.13.1805,Geometry & Topology,Ilya Kapovich,Martin Lust
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Geometric intersection number and analogues of the curve complex for free groups
(
Citations: 9
)
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Ilya Kapovich
,
Martin Lustig
Journal:
Geometry & Topology  GEOM TOPOL
, vol. 13, no. 3, pp. 18051833, 2009
DOI:
10.2140/gt.2009.13.1805
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www.msp.warwick.ac.uk
)
Citation Context
(3)
...The intersection form was introduced in [Ka3,4], [Lu] for free simplicial actions of F , that is for the nonprojectivized outer space cv(F ). In a recent paper [
KaLu2
] we proved that the intersection form extends continuously to the closure cv(F )o fcv(F ) consisting of all minimal very small isometric actions of F on Rtrees...
...Ursula Hamenst¨ adt [H] recently used our result from [
KaLu2
] about the continuous extension of the intersection form to cv(F ) as a key ingredient to prove that any nonelementary subgroup of Out(F ), where N ≥ 3, has infinitedimensional second bounded cohomology group (infinitedimensional space of quasimorphisms)...
...Very recently Bestvina and Feighn [BeF2] used [
KaLu2
] to show that for any finite collection φ1 ,...,φ m ∈ Out(FN )o fiwip outer automorphisms of FN (“irreducible automorphisms with irreducible powers”, see Definition 12.1) there exists a δhyperbolic complex X = X(φ1 ,...,φ m )w ith an isometric Out( FN )action where each φi acts with a positive asymptotic translation length...
...If Tλ denotes the “dual” Rtree transverse to λ with metric defined by the transverse measure on λ (see Ch. 11.12 in [Kap] for details), then the definition in [
KaLu2
] gives...
...One of the main motivations and prospective uses for Theorem 1.1 is to analyze the intersection graph, introduced by the authors in [
KaLu2
] in order to study various free group analogues of the curve complex...
...Thus one can define a graph, whose vertices are conjugacy classes of primitive elements in F where two vertices [a], [b ]a re adjacent if there existsT ∈ cv(F ) such that � T, ηa� = � T, ηb� =0 , that is � a� T = � b� T = 0 (this graph is almost the same as the “dual cut graph” defined in [
KaLu2
])...
...This leads to the notion [
KaLu2
] of a cut graph for F whose vertices are nontrivial splittings of F as the fundamental group of a graph of groups with a single edge and the trivial edge group, and where adjacency again corresponds to having a common refinement...
...In [
KaLu2
] we prove that for N ≥ 3 the intersection graph and all the free group analogues of the curve complex derived from it have infinite diameter, by analyzing the action of iwip automorphisms (see Definition 12.1)...
...In the terminology of [
KaLu2
], the assumption of Theorem 1.4 says that the distance between [T1 ]a nd [ T2] in the intersection graph I(F ) is bigger than two...
...Recently, Kapovich and Lustig [
KaLu2
] generalized this result to the case of arbitrary very small actions and proved Proposition–Definition 2.11 in the form stated above...
...Therefore, by the continuity of the intersection form [
KaLu2
], we have...
...By the continuity of the intersection form on the closure of the nonprojectivized outer space (see [
KaLu2
]), this implies that...
Ilya Kapovich
,
et al.
Intersection Form, Laminations and Currents on Free Groups
...A discussion of some of these analogs and their basic properties is provided in [
23
]...
...By analogy with the curve complex situation where pseudoAnosov homeomorphisms have unbounded orbits and other homeomorphisms have bounded orbits, Kapovich and Lustig have shown that fully irreducible automorphisms act with unbounded orbits and other automorphisms with bounded orbits [
23
]...
Jason Behrstock
,
et al.
Growth of intersection numbers for free group automorphisms
...It should be remarked that different notions of intersection number have been developed by Scott–Swarup [SS], Guirardel [Gu2], and Kapovich–Lustig [
KL3
], but that ours has been tailored to suit the needs of our theorem...
...To build filling cyclic trees for arbitrarily high rank free groups we introduce two simplicial complexes naturally associated to Fk; these complexes appear in [
KL3
]...
...is the following variant of the likenamed complex appearing in [
KL3
]: Vertices correspond to very small simplicial trees for Fk, i.e...
...Remark 2.9. For k ≥ 3, Kapovich and Lustig have shown that for a hyperbolic fully irreducible element φ ∈ Out Fk and any two vertices [A], [B] ∈D that dD([A] ,φ n ([B])) goes to infinity as n →± ∞([
KL3
])...
Matt Clay
,
et al.
Twisting Out Fully Irreducible Automorphisms
References
(24)
The topology at infinity of Out(Fn)
(
Citations: 29
)
Mladen Bestvina
,
Mark Feighn
Journal:
Inventiones Mathematicae  INVENT MATH
, vol. 140, no. 3, pp. 651692, 2000
Geometry of the complex of curves I: Hyperbolicity
(
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Howard A. Masur
,
Yair N. Minsky
Journal:
Inventiones Mathematicae  INVENT MATH
, vol. 138, no. 1, pp. 103149, 1999
3Manifolds as viewed from the curve complex
(
Citations: 78
)
John Hempel
Journal:
Topology
, vol. 40, no. 3, pp. 631657, 2001
Very small group actions on Rtrees and dehn twist automorphisms
(
Citations: 30
)
Marshall M. Cohen
,
Martin Lustig
Journal:
Topology
, vol. 34, no. 3, pp. 575617, 1995
Dynamics of Out( F n ) on the boundary of outer space
(
Citations: 9
)
Vincent Guirardel
Journal:
Annales Scientifiques De L Ecole Normale Superieure  ANN SCI ECOLE NORM SUPER
, vol. 33, no. 4, pp. 433465, 2000
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Citations
(9)
Intersection Form, Laminations and Currents on Free Groups
(
Citations: 11
)
Ilya Kapovich
,
Martin Lustig
Journal:
Geometric and Functional Analysis  GEOM FUNCT ANAL
, vol. 19, no. 5, pp. 14261467, 2010
Growth of intersection numbers for free group automorphisms
(
Citations: 2
)
Jason Behrstock
,
Mladen Bestvina
,
Matt Clay
Journal:
Journal of Topology  J TOPOL
, vol. 3, no. 2, pp. 280310, 2010
Twisting Out Fully Irreducible Automorphisms
(
Citations: 1
)
Matt Clay
,
Alexandra Pettet
Journal:
Geometric and Functional Analysis  GEOM FUNCT ANAL
, vol. 20, no. 3, pp. 657689, 2010
A hyperbolic Out(Fn)complex
(
Citations: 1
)
Mladen Bestvina
,
Mark Feighn
Journal:
Groups Geometry and Dynamics  GROUP GEOM DYN
, pp. 3158, 2010
The complex of partial bases for F_n and finite generation of the Torelli subgroup of Aut(F_n)
Matthew B. Day
,
Andrew Putman
Published in 2010.