## Invariable generation of certain groups of piecewise linear homeomorphisms of the interval

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- by Yoshifumi Matsuda and Shigenori Matsumoto
- Proc. Amer. Math. Soc.
**149**(2021), 1-11 - DOI: https://doi.org/10.1090/proc/15277
- Published electronically: October 9, 2020
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## Abstract:

Let $P$ be the group of all the orientation preserving piecewise linear homeomorphisms of the interval $[0,1]$. Given any $a>1$, let $P^a$ be the subgroup of $P$ consisting of all the elements with slopes in $a^\mathbb {Z}$, and let $P^\mathbb {Q}$ be the subgroup of $P$ consisting of all the elements with slopes and breaks in $\mathbb {Q}$. We show that the groups $P$, $P^a$, $P^\mathbb {Q}$, as well as Thompson group $F$, are invariably generated.## References

- J. M. Belk,
*Thompson’s group $F$*, Thesis, Cornell University. - Mladen Bestvina and Koji Fujiwara,
*Handlebody subgroups in a mapping class group*, In the tradition of Ahlfors-Bers. VII, Contemp. Math., vol. 696, Amer. Math. Soc., Providence, RI, 2017, pp. 29–50. MR**3715440**, DOI 10.1090/conm/696/14015 - J. W. Cannon, W. J. Floyd, and W. R. Parry,
*Introductory notes on Richard Thompson’s groups*, Enseign. Math. (2)**42**(1996), no. 3-4, 215–256. MR**1426438** - Tsachik Gelander,
*Convergence groups are not invariably generated*, Int. Math. Res. Not. IMRN**19**(2015), 9806–9814. MR**3431613**, DOI 10.1093/imrn/rnu248 - Tsachik Gelander, Gili Golan, and Kate Juschenko,
*Invariable generation of Thompson groups*, J. Algebra**478**(2017), 261–270. MR**3621672**, DOI 10.1016/j.jalgebra.2017.01.019 - R. I. Grigorčuk,
*On Burnside’s problem on periodic groups*, Funktsional. Anal. i Prilozhen.**14**(1980), no. 1, 53–54 (Russian). MR**565099** - Graham Higman, B. H. Neumann, and Hanna Neumann,
*Embedding theorems for groups*, J. London Math. Soc.**24**(1949), 247–254. MR**32641**, DOI 10.1112/jlms/s1-24.4.247 - William M. Kantor, Alexander Lubotzky, and Aner Shalev,
*Invariable generation of infinite groups*, J. Algebra**421**(2015), 296–310. MR**3272383**, DOI 10.1016/j.jalgebra.2014.08.030 - A. Yu. Ol′shanskiĭ,
*Groups of bounded period with subgroups of prime order*, Algebra i Logika**21**(1982), no. 5, 553–618 (Russian). MR**721048** - James Wiegold,
*Transitive groups with fixed-point free permutations*, Arch. Math. (Basel)**27**(1976), no. 5, 473–475. MR**417300**, DOI 10.1007/BF01224701 - James Wiegold,
*Transitive groups with fixed-point-free permutations. II*, Arch. Math. (Basel)**29**(1977), no. 6, 571–573. MR**463299**, DOI 10.1007/BF01220455

## Bibliographic Information

**Yoshifumi Matsuda**- Affiliation: Department of Physics and Mathematics, College of Science and Engineering, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara-shi, Kanagawa, 252-5258 Japan
- MR Author ID: 878267
- ORCID: 0000-0001-9092-4255
- Email: ymatsuda@gem.aoyama.ac.jp
**Shigenori Matsumoto**- Affiliation: Department of Mathematics, College of Science and Technology, Nihon University, 1-8-14 Kanda-Surugadai, Chiyoda-ku, Tokyo, 101-8308 Japan
- MR Author ID: 214791
- ORCID: 0000-0002-5851-7235
- Email: matsumo@math.cst.nihon-u.ac.jp
- Received by editor(s): August 18, 2018
- Received by editor(s) in revised form: September 4, 2018, September 20, 2019, and March 11, 2020
- Published electronically: October 9, 2020
- Additional Notes: The first author was partially supported by Grant-in-Aid for Scientific Research (C) No. 17K05260.

The second author was partially supported by Grant-in-Aid for Scientific Research (C) No. 18K03312. - Communicated by: Kenneth Bromberg
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 1-11 - MSC (2010): Primary 20F65; Secondary 20F05
- DOI: https://doi.org/10.1090/proc/15277
- MathSciNet review: 4172581