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Relative free splitting and free factor complexes: An overview

  • Michael Handel
  • Lee Mosher
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Abstract

For any group ฮ“ and any free factor system ๐’œ of ฮ“, the relative outer automorphism group Out(ฮ“;๐’œ) acts naturally on the relative free splitting complex โ„ฑ\!๐’ฎ(ฮ“;๐’œ) and on the complex of relative free factor systems โ„ฑ\!โ„ฑ(ฮ“;๐’œ), generalizing the well known actions of Out(Fn) on the absolute free splitting complex โ„ฑ\!๐’ฎ(Fn) and the absolute free factor complex โ„ฑ(Fn) of the rank n free group Fn. This overview summarizes a three part work regarding the large scale geometry of โ„ฑ\!๐’ฎ(ฮ“;๐’œ) and โ„ฑ\!โ„ฑ(ฮ“;๐’œ) and the geometric dynamics of the actions on these complexes by elements of Out(ฮ“;๐’œ). In Part I arXiv:1407.3508 we prove hyperbolicity of โ„ฑ\!๐’ฎ(ฮ“;๐’œ) and of โ„ฑ\!โ„ฑ(ฮ“;๐’œ). In Parts II and III arXiv:2212.09907, arXiv:2503.07532 we study the relation between the geometric dynamics of an element of Out(ฮ“;๐’œ) and the dynamics of its relative train track representatives. The main tool in Part II is the Two Over All Theorem, expressing an exponential flaring property of Stallings fold paths in โ„ฑ\!๐’ฎ(ฮ“;๐’œ). The main tools in Part III are "filling paths", used to formulate and prove a strong version of the Two Over All Theorem.

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Cite

BibTeXRIS
BibTeX
@misc{handel2026relative,
  title = {{Relative free splitting and free factor complexes: An overview}},
  author = {Michael Handel and Lee Mosher},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.19249},
  eprint = {2607.19249},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.19249},
}
RIS
TY  - UNPB
TI  - Relative free splitting and free factor complexes: An overview
AU  - Michael Handel
AU  - Lee Mosher
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2607.19249
UR  - https://arxiv.org/abs/2607.19249
AB  - For any group ฮ“ and any free factor system ๐’œ of ฮ“, the relative outer automorphism group Out(ฮ“;๐’œ) acts naturally on the relative free splitting complex โ„ฑ\!๐’ฎ(ฮ“;๐’œ) and on the complex of relative free factor systems โ„ฑ\!โ„ฑ(ฮ“;๐’œ), generalizing the well known actions of Out(F_n) on the absolute free splitting complex โ„ฑ\!๐’ฎ(F_n) and the absolute free factor complex โ„ฑ(F_n) of the rank n free group F_n. This overview summarizes a three part work regarding the large scale geometry of โ„ฑ\!๐’ฎ(ฮ“;๐’œ) and โ„ฑ\!โ„ฑ(ฮ“;๐’œ) and the geometric dynamics of the actions on these complexes by elements of Out(ฮ“;๐’œ). In Part I arXiv:1407.3508 we prove hyperbolicity of โ„ฑ\!๐’ฎ(ฮ“;๐’œ) and of โ„ฑ\!โ„ฑ(ฮ“;๐’œ). In Parts II and III arXiv:2212.09907, arXiv:2503.07532 we study the relation between the geometric dynamics of an element of Out(ฮ“;๐’œ) and the dynamics of its relative train track representatives. The main tool in Part II is the Two Over All Theorem, expressing an exponential flaring property of Stallings fold paths in โ„ฑ\!๐’ฎ(ฮ“;๐’œ). The main tools in Part III are "filling paths", used to formulate and prove a strong version of the Two Over All Theorem.
ER  -