| Title: |
Cannon-Thurston fibers for iwip automorphisms of $F_N$ |
| Authors: |
Kapovich, Ilya; Lustig, Martin |
| Contributors: |
Department of Mathematics Urbana; University of Illinois at Urbana-Champaign Urbana (UIUC); University of Illinois System-University of Illinois System; Institut de Mathématiques de Marseille (I2M); Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS); The first author was partially supported by the NSF grants DMS-0904200 and DMS-0404991, and by the Simons Foundation grant no. 279836. Both authors acknowledge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 ‘RNMS: GEometric structures And Representation varieties’ (the GEAR Network). |
| Source: |
ISSN: 0024-6107. |
| Publisher Information: |
CCSD; London Mathematical Society; Wiley |
| Publication Year: |
2015 |
| Collection: |
Aix-Marseille Université: HAL |
| Subject Terms: |
20F65; 57M; 37B; 37D; [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] |
| Description: |
22 pages ; International audience ; For any atoroidal iwip $\phi \in Out(F_N)$ the mapping torus group $G_\phi=F_N\rtimes_\phi e$ is hyperbolic, and the embedding $\iota: F_N \overset{\lhd}{\longrightarrow} G_\phi$ induces a continuous, $F_N$-equivariant and surjective {\em Cannon-Thurston map} $\hat \iota: \partial F_N \to \partial G_\phi$. We prove that for any $\phi$ as above, the map $\hat \iota$ is finite-to-one and that the preimage of every point of $\partial G_\phi$ has cardinality $\le 2N$. We also prove that every point $S\in \partial G_\phi$ with $\ge 3$ preimages in $\partial F_N$ has the form $(wt^m)^\infty$ where $w\in F_N, m\ne 0$, and that there are at most $4N-5$ distinct $F_N$-orbits of such {\em singular} points in $\partial G_\phi$ (for the translation action of $F_N$ on $\partial G_\phi$). By contrast, we show that for $k=1,2$ there are uncountably many points $S\in \partial G_\phi$ (and thus uncountably many $F_N$-orbits of such $S$) with exactly $k$ preimages in $\partial F_N$. |
| Document Type: |
article in journal/newspaper |
| Language: |
English |
| Relation: |
info:eu-repo/semantics/altIdentifier/arxiv/1207.3494; ARXIV: 1207.3494 |
| DOI: |
10.1112/jlms/jdu069 |
| Availability: |
https://hal.science/hal-01318434; https://doi.org/10.1112/jlms/jdu069 |
| Accession Number: |
edsbas.754246B6 |
| Database: |
BASE |