Subgroup Decomposition in Mathrm{Out}(F_{n})

Subgroup Decomposition in Mathrm{Out}(F_{n})

In this work the authors develop a decomposition theory for subgroups of \mathsf{Out}(F_n) which generalizes the decomposition theory for individual elements of \mathsf{Out}(F_n) found in the work of Bestvina, Feighn, and Handel, and which is analogous to the decomposition theory for subgroups of ma...

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Bibliographic Details
Main Author: Handel, Michael
Other Authors: Mosher, Lee
Format: eBook
Language:English
Published: Providence : American Mathematical Society, 2020.
Series:Memoirs of the American Mathematical Society Ser.
Subjects:
Non-Abelian groups.
Geometric group theory.
Decomposition (Mathematics)
Automorphisms.
Algebraic topology.
Algebraic topology
Automorphisms
Geometric group theory
Non-Abelian groups
Manifolds and cell complexes {For complex manifolds, see 32Qxx} > Low-dimensional topology > Topological methods in group theory.
Group theory and generalizations > Structure and classification of infinite or finite groups > Free nonabelian groups.
Group theory and generalizations > Special aspects of infinite or finite groups > Geometric group theory [See also 05C25, 20E08, 57Mxx].
Group theory and generalizations > Special aspects of infinite or finite groups > Automorphism groups of groups [See also 20E36].
Online Access:Click for online access
Table of Contents:
  • Cover
  • Title page
  • Introduction to Subgroup Decomposition
  • The main theorem, slightly simplified
  • Rotationless versus \IA_{ }(\Z/3) (\PartTwo)
  • The main theorem, full version
  • The relative Kolchin theorem for \Out( _{ }) (\PartTwo)
  • Geometric models (\PartOne)
  • Vertex group systems (\PartOne)
  • Weak attraction theory (\PartThree)
  • Relatively irreducible subgroups (\PartFour)
  • Part I . Geometric Models
  • Introduction to \PartOne
  • Chapter 1. Preliminaries: Decomposing outer automorphisms
  • 1.1. _{ } and its subgroups, marked graphs, and lines
  • 1.1.1. The geometry of _{ } and its subgroups.
  • 1.1.2. Subgroup systems, free factor systems, and malnormality.
  • Malnormal subgroup systems.
  • 1.1.3. Restrictions of outer automorphisms.
  • 1.1.4. Marked graphs, paths, and circuits.
  • 1.1.5. Spaces of paths and lines.
  • Rays.
  • 1.1.6. The path maps _{#} and _{##}, and bounded cancellation.
  • 1.2. Subgroup systems carrying lines and other things
  • 1.2.1. Subgroup systems carrying lines, rays, and conjugacy classes.
  • 1.2.2. Free factor supports of lines, rays, and subgroup systems.
  • Remark.
  • 1.3. Attracting laminations
  • Remarks.
  • 1.4. Principal automorphisms and rotationless outer automorphisms
  • Remark.
  • 1.5. Relative train track maps and \ct s
  • 1.5.1. Definitions of relative train track maps and \cts
  • Topological representatives and Nielsen paths.
  • Filtrations and height.
  • Directions and turns.
  • Enveloping of zero strata.
  • 1.5.2. Facts about \cts, their Nielsen paths, and their zero strata.
  • Remark.
  • Zero strata.
  • 1.5.3. Facts about principal lifts, principal directions, and principal rays.
  • Uniqueness of principal rays.
  • Weak accumulation of attracting fixed points.
  • 1.5.4. Properties of \eg strata.
  • 1.6. Properties of Attracting Laminations
  • 1.6.1. The relation between \eg strata and attracting laminations.
  • 1.6.2. Tiles and their applications.
  • Characterizing attracting laminations.
  • Weak attraction of paths and circuits.
  • Principal rays.
  • Generic leaves.
  • Construction of an attracting neighborhood basis.
  • 1.6.3. \eg principal rays and \Fix_{ }
  • 1.6.4. Pushing forward attracting laminations.
  • Chapter 2. Geometric \eg strata and geometric laminations
  • 2.1. Defining and characterizing geometric strata
  • 2.1.1. Defining weak geometric models and geometric strata.
  • Remark: Comparing Definition 2.2 to Definition 5.1.4 of \BookOne.
  • 2.1.2. Defining geometric models.
  • 2.1.3. Invariant free factor systems associated to a geometric model.
  • 2.1.4. Characterizing geometric strata: Proof of Fact 2.3
  • Remark.
  • Remark.
  • 2.2. Complementary subgraph and peripheral splitting
  • 2.3. The laminations of a geometric stratum
  • 2.3.1. Review of Nielsen-Thurston theory.
  • Remarks on the proof.
  • 2.3.2. Comparing free group laminations and surface laminations.

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