Canonical reducing curves of surface homeomorphism

Wu Yingqing

doi:10.1007/bf02559911

Out(F3) Index Realization

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004115000456 ◽

2015 ◽

Vol 159 (3) ◽

pp. 445-458 ◽

Cited By ~ 1

Author(s):

CATHERINE PFAFF

Keyword(s):

Free Group ◽

Quadratic Differentials ◽

Singularity Index ◽

Outer Automorphisms ◽

Teichmüller Flow ◽

Surface Homeomorphism

AbstractBy proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie [MS93] determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this paper we determine an analog to the theorem forOut(F3). That is, we determine which index lists permitted by the [GJLL98] index sum inequality are achieved by ageometric fully irreducible outer automorphisms of the rank-3 free group.

Lefschetz numbers of periodic orbits of pseudo-Anosov homeomorphisms

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100071966 ◽

1994 ◽

Vol 115 (1) ◽

pp. 121-132 ◽

Cited By ~ 1

Author(s):

John Guaschi

Keyword(s):

Periodic Orbit ◽

Periodic Orbits ◽

Periodic Points ◽

Lefschetz Numbers ◽

Finite Set ◽

Surface Homeomorphism

AbstractGiven a surface homeomorphism isotopic to the identity which is pseudo-Anosov relative to a finite set, we show that the sum of the Lefschetz numbers of periodic points of any period greater than one is non-negative. If this period is odd and greater than a number which depends only on the surface, the sum is zero. If we consider sequences of periods such that each element is twice that of its predecessor, then this sum is increasing beyond a certain point also depending on the surface. As a corollary, for each periodic orbit contained within the boundary of the surface there exists one of the same period contained in the interior.

Congruence Classes of Orientable $2$-Cell Embeddings of Bouquets of Circles and Dipoles

The Electronic Journal of Combinatorics ◽

10.37236/313 ◽

2010 ◽

Vol 17 (1) ◽

Cited By ~ 2

Author(s):

Yan-Quan Feng ◽

Jin-Ho Kwak ◽

Jin-Xin Zhou

Keyword(s):

Orientable Surface ◽

Simple Graph ◽

Complete Graphs ◽

Graph Automorphism ◽

Wheel Graphs ◽

Complete Bipartite Graphs ◽

Multiple Edges ◽

Orientable Surfaces ◽

Surface Homeomorphism ◽

Congruence Classes

Two $2$-cell embeddings $\imath : X \to S$ and $\jmath : X \to S$ of a connected graph $X$ into a closed orientable surface $S$ are congruent if there are an orientation-preserving surface homeomorphism $h : S \to S$ and a graph automorphism $\gamma$ of $X$ such that $\imath h =\gamma\jmath$. Mull et al. [Proc. Amer. Math. Soc. 103(1988) 321–330] developed an approach for enumerating the congruence classes of $2$-cell embeddings of a simple graph (without loops and multiple edges) into closed orientable surfaces and as an application, two formulae of such enumeration were given for complete graphs and wheel graphs. The approach was further developed by Mull [J. Graph Theory 30(1999) 77–90] to obtain a formula for enumerating the congruence classes of $2$-cell embeddings of complete bipartite graphs into closed orientable surfaces. By considering automorphisms of a graph as permutations on its dart set, in this paper Mull et al.'s approach is generalized to any graph with loops or multiple edges, and by using this method we enumerate the congruence classes of $2$-cell embeddings of a bouquet of circles and a dipole into closed orientable surfaces.

Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

Journal of Topology and Analysis ◽

10.1142/s1793525316500084 ◽

2016 ◽

Vol 08 (02) ◽

pp. 207-242 ◽

Cited By ~ 1

Author(s):

Catherine Pfaff

Keyword(s):

Direct Consequence ◽

Quadratic Differentials ◽

Text Setting ◽

Singularity Index ◽

Outer Automorphisms ◽

Analogous Formula ◽

Surface Homeomorphisms ◽

Rank 2 ◽

Surface Homeomorphism ◽

The Ideal

By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie [24] determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to an [Formula: see text] analog of the Masur–Smillie theorem. Since the ideal Whitehead graphs defined by Handel and Mosher [16] give a strictly finer invariant in the analogous [Formula: see text] setting, we determine which of the 21 connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible outer automorphisms in [Formula: see text]. The rank 2 case is actually a direct consequence of the work of Masur and Smillie, as all elements of [Formula: see text] are induced by surface homeomorphisms and the index list and ideal Whitehead graph for a surface homeomorphism give precisely the same data.