## The equation $x^py^q=z^r$ and groups that act freely on $\Lambda$-trees

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- by N. Brady, L. Ciobanu, A. Martino and S. O Rourke PDF
- Trans. Amer. Math. Soc.
**361**(2009), 223-236 Request permission

## Abstract:

Let $G$ be a group that acts freely on a $\Lambda$-tree, where $\Lambda$ is an ordered abelian group, and let $x, y, z$ be elements in $G$. We show that if $x^p y^q = z^r$ with integers $p$, $q$, $r \geq 4$, then $x$, $y$ and $z$ commute. As a result, the one-relator groups with $x^p y^q = z^r$ as relator, are examples of hyperbolic and CAT($-1$) groups which do not act freely on any $\Lambda$-tree.## References

- Hyman Bass,
*Group actions on non-Archimedean trees*, Arboreal group theory (Berkeley, CA, 1988) Math. Sci. Res. Inst. Publ., vol. 19, Springer, New York, 1991, pp. 69–131. MR**1105330**, DOI 10.1007/978-1-4612-3142-4_{3} - Benjamin Baumslag,
*Residually free groups*, Proc. London Math. Soc. (3)**17**(1967), 402–418. MR**215903**, DOI 10.1112/plms/s3-17.3.402 - Gilbert Baumslag,
*On generalised free products*, Math. Z.**78**(1962), 423–438. MR**140562**, DOI 10.1007/BF01195185 - Gilbert Baumslag,
*On a problem of Lyndon*, J. London Math. Soc.**35**(1960), 30–32. MR**111780**, DOI 10.1112/jlms/s1-35.1.30 - Ian Chiswell,
*Introduction to $\Lambda$-trees*, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR**1851337**, DOI 10.1142/4495 - I. M. Chiswell,
*Some examples of groups with no nontrivial action on a $\Lambda$-tree*, Mathematika**42**(1995), no. 1, 214–219. MR**1346688**, DOI 10.1112/S0025579300011517 - Warren Dicks and H. H. Glover,
*An algorithm for cellular maps of closed surfaces*, Enseign. Math. (2)**43**(1997), no. 3-4, 207–252. MR**1489884** - A. M. Gaglione and D. Spellman,
*Generalizations of free groups: some questions*, Comm. Algebra**22**(1994), no. 8, 3159–3169. MR**1272379**, DOI 10.1080/00927879408825019 - S. M. Gersten and H. B. Short,
*Small cancellation theory and automatic groups*, Invent. Math.**102**(1990), no. 2, 305–334. MR**1074477**, DOI 10.1007/BF01233430 - Vincent Guirardel,
*Limit groups and groups acting freely on $\Bbb R^n$-trees*, Geom. Topol.**8**(2004), 1427–1470. MR**2119301**, DOI 10.2140/gt.2004.8.1427 - T. Hsu and D. Wise. Cubulating graphs of free groups with cyclic edge groups. In preparation, 2006.
- O. Kharlampovich and A. Myasnikov,
*Irreducible affine varieties over a free group. I. Irreducibility of quadratic equations and Nullstellensatz*, J. Algebra**200**(1998), no. 2, 472–516. MR**1610660**, DOI 10.1006/jabr.1997.7183 - R. C. Lyndon,
*The equation $a^{2}b^{2}=c^{2}$ in free groups*, Michigan Math. J.**6**(1959), 89–95. MR**103218** - R. C. Lyndon and M. P. Schützenberger,
*The equation $a^{M}=b^{N}c^{P}$ in a free group*, Michigan Math. J.**9**(1962), 289–298. MR**162838** - A. Martino and S. O’Rourke,
*Free actions on $\Bbb Z^n$-trees: a survey*, Geometric methods in group theory, Contemp. Math., vol. 372, Amer. Math. Soc., Providence, RI, 2005, pp. 11–25. MR**2139673**, DOI 10.1090/conm/372/06870 - A. Martino and S. O Rourke,
*Some free actions on non-Archimedean trees*, J. Group Theory**7**(2004), no. 2, 275–286. MR**2049022**, DOI 10.1515/jgth.2004.009 - V. N. Remeslennikov,
*$\exists$-free groups*, Sibirsk. Mat. Zh.**30**(1989), no. 6, 193–197 (Russian); English transl., Siberian Math. J.**30**(1989), no. 6, 998–1001 (1990). MR**1043446**, DOI 10.1007/BF00970922 - Eugene Schenkman,
*The equation $a^{n}b^{n}=c^{n}$ in a free group*, Ann. of Math. (2)**70**(1959), 562–564. MR**104723**, DOI 10.2307/1970329 - Marcel-Paul Schützenberger,
*Sur l’équation $a^{2+n}=b^{2+m}c^{2+p}$ dans un groupe libre*, C. R. Acad. Sci. Paris**248**(1959), 2435–2436 (French). MR**103219** - Zlil Sela,
*Diophantine geometry over groups. I. Makanin-Razborov diagrams*, Publ. Math. Inst. Hautes Études Sci.**93**(2001), 31–105. MR**1863735**, DOI 10.1007/s10240-001-8188-y

## Additional Information

**N. Brady**- Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
- Email: nbrady@math.ou.edu
**L. Ciobanu**- Affiliation: Department of Mathematics, University of Fribourg, CH-1700 Fribourg, Switzerland
- MR Author ID: 797163
- Email: laura.ciobanu@unifr.ch
**A. Martino**- Affiliation: Department of Mathematics, Universitat Politècnica de Catalunya, 08860 Castelldefels, Spain
- MR Author ID: 646503
- Email: Armando.Martino@upc.edu
**S. O Rourke**- Affiliation: Department of Mathematics, Cork Institute of Technology, Cork, Ireland
- Email: Shane.ORourke@cit.ie
- Received by editor(s): December 6, 2006
- Published electronically: August 19, 2008
- © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**361**(2009), 223-236 - MSC (2000): Primary 20E08, 20F65
- DOI: https://doi.org/10.1090/S0002-9947-08-04639-4
- MathSciNet review: 2439405