## Algebraic & Geometric Topology

### Highly transitive actions of free products

#### Abstract

We characterize free products admitting a faithful and highly transitive action. In particular, we show that the group $PSL2(ℤ)≃(ℤ∕2ℤ)∗(ℤ∕3ℤ)$ admits a faithful and highly transitive action on a countable set.

#### Article information

Source
Algebr. Geom. Topol., Volume 13, Number 1 (2013), 589-607.

Dates
Revised: 16 October 2012
Accepted: 5 July 2012
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715508

Digital Object Identifier
doi:10.2140/agt.2013.13.589

Mathematical Reviews number (MathSciNet)
MR3116381

Zentralblatt MATH identifier
1292.20005

#### Citation

Moon, Soyoung; Stalder, Yves. Highly transitive actions of free products. Algebr. Geom. Topol. 13 (2013), no. 1, 589--607. doi:10.2140/agt.2013.13.589. https://projecteuclid.org/euclid.agt/1513715508

#### References

• M Bestvina, $\mathbb{R}$–trees in topology, geometry, and group theory, from: “Handbook of geometric topology”, (R J Daverman, R B Sher, editors), North-Holland, Amsterdam (2002) 55–91
• J D Dixon, Most finitely generated permutation groups are free, Bull. London Math. Soc. 22 (1990) 222–226
• S Garion, Y Glasner, Highly transitive actions of $\mathrm{Out}(\mathbb{F}_n)$, to appear in Groups, Geometry and Dynamics (2011)
• A M W Glass, S H McCleary, Highly transitive representations of free groups and free products, Bull. Austral. Math. Soc. 43 (1991) 19–36
• S V Gunhouse, Highly transitive representations of free products on the natural numbers, Arch. Math. (Basel) 58 (1992) 435–443
• K K Hickin, Highly transitive Jordan representations of free products, J. London Math. Soc. (2) 46 (1992) 81–91
• D Kitroser, Highly-transitive actions of surface groups, Proc. Amer. Math. Soc. 140 (2012) 3365–3375
• S Moon, Permanence properties of amenable, transitive and faithful actions, Bull. Belg. Math. Soc. Simon Stevin 18 (2011) 287–296
• B H Neumann, Groups covered by permutable subsets, J. London Math. Soc. 29 (1954) 236–248
• P M Neumann, The structure of finitary permutation groups, Arch. Math. $($Basel$)$ 27 (1976) 3–17
• J-P Serre, Arbres, amalgames, ${\rm SL}\sb{2}$, Astérisque 46, Société Mathématique de France, Paris (1977)