By Michael Handel, Lee Mosher

The authors strengthen a suggestion of axis within the Culler-Vogtmann outer house $\mathcal{X}_r$ of a finite rank loose crew $F_r$, with recognize to the motion of a nongeometric, absolutely irreducible outer automorphism $\phi$. not like the location of a loxodromic isometry performing on hyperbolic area, or a pseudo-Anosov mapping classification performing on Teichmuller house, $\mathcal{X}_r$ has no usual metric, and $\phi$ turns out to not have a unmarried average axis. in its place those axes for $\phi$, whereas now not distinct, healthy into an ""axis bundle"" $\mathcal{A}_\phi$ with great topological homes: $\mathcal{A}_\phi$ is a closed subset of $\mathcal{X}_r$ right homotopy resembling a line, it really is invariant less than $\phi$, the 2 ends of $\mathcal{A}_\phi$ restrict at the repeller and attractor of the source-sink motion of $\phi$ on compactified outer house, and $\mathcal{A}_\phi$ relies clearly at the repeller and attractor.

The authors suggest quite a few definitions for $\mathcal{A}_\phi$, each one prompted in numerous methods by way of educate tune idea or through houses of axes in Teichmuller house, and so they end up their equivalence.

Read or Download Axes in outer space PDF

Best science & mathematics books

The Mathematical Works of Bernard Bolzano

Bolzano's (1781-1848) pursuits reflected these of the nineteenth century, together with theology, philosophy, ethics, politics, good judgment and arithmetic. His paintings used to be all of a bit, with the intention to examine one self-discipline calls for learning his entire works. as a way to make extra of Bolzano's paintings to be had in English, this new translation bargains a consultant number of the mathematical paintings.

Programmieren mit Wolfram Mathematica®

Dieses von einem erfahrenen Praktiker geschriebene Buch wendet sich an alle Benutzer von Mathematica®. Der Neuling macht schnell die ersten Schritte und lernt vor allem zunächst, die eingebaute Hilfe und Dokumentation effizient zu nutzen. Dem Fortgeschrittenen helfen die prototypischen Beispiele und vor allem auch eine umfangreiche Sammlung elementarer methods und Kniffe.

Haphazard reality: Half a century of science

Casimir, himself a recognized medical professional, studied and labored with 3 nice physicists of the 20 th century: Niels Bohr, Wolfgang Pauli and Paul Ehrenfest. In his autobiography, the bright theoretician we could the reader witness the revolution that resulted in quantum physics, whose impression on smooth society became out to be repeatedly higher than the 1st atomic physicists may have imagined.

Der Aufstieg der Wissenschaftlichen Philosophie

ZUR 2. AUFLAGE DER DEUTSCHEN AUSGABE Warum eröffnen wir eine Buchreihe über Wissenschafts­ theorie verschiedener Hauptdisziplinen mit einer Neu­ auflage von Hans Reichenbachs Buch "Aufstieg der wissen­ schaftlichen Philosophie", dessen 1. Auflage 1951 in den united states erschien? Weil es sich mit seiner präzisen und ver­ ständlichen Sprache nach siebzehn Jahren immer noch so frisch wie damals liest, weil seine Problematik zwischen Rationalismus und Empirismus, seine competition gegen Metaphysik und Ontologie als philosophische Grund­ legung der Naturwissenschaft immer noch in Mittel­ europa aktuell ist.

Extra resources for Axes in outer space

Sample text

If φ is rotationless then φ is geometric if and only if φ is represented by a principle automorphism with nontrivial ﬁxed subgroup. Let g : G → G be an irreducible train track representative of φ. A lift g˘ : G → G is a principal lift if the corresponding automorphism is a principal automorphism, and so two principal lifts g˘, g˘ are equivalent if and only if g˘ = tγ ◦ g˘ ◦ t−1 γ for some γ ∈ Fr . Consider a periodic vertex v ∈ G. Since v has at least one legal turn, v has at least two periodic directions.

A path σ is a periodic Nielsen path if g# minimal p ≥ 1 called the period of σ. A Nielsen path is a periodic Nielsen path with period one. A periodic Nielsen path is indivisible if it can not be written as a non-trivial concatenation of periodic Nielsen subpaths. A path σ is a periodic k (σ) is a periodic pre-Nielsen path if it is not a periodic Nielsen path but some g# Nielsen path. We record information about Nielsen paths in the following lemma. 7. Assume that g : G → G is an irreducible train track map.

The lift ﬁxed, vertex. There is a path σ connecting aj to c so that g# σ ˜ with endpoint a ˜j terminates at a principal vertex c˜ and fg (˜ c) = fg (˜ aj ) = b. This completes the proof of (4) and so the proof of the lemma. 4 of [BFH97], and whose second items follows from the elementary fact that the translation lengths of nontrivial elements of a ﬁnite rank free group acting discretely on an R-tree are bounded away from zero. 17. For each nongeometric, fully irreducible φ ∈ Out(Fr ) and each nontrivial subgroup A < Fr of inﬁnite index, the following hold: (1) The action of A on T+φ is discrete.

Download PDF sample

Rated 4.79 of 5 – based on 21 votes