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

Author(s): N. Brady; L. Ciobanu; A. Martino; S. O Rourke
Journal: Trans. Amer. Math. Soc. 361 (2009), 223-236.
MSC (2000): Primary 20E08, 20F65
Posted: August 19, 2008
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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:

1.
Hyman Bass.
Group actions on non-Archimedean trees.
In Arboreal group theory (Berkeley, CA, 1988), volume 19 of Math. Sci. Res. Inst. Publ., pages 69-131. Springer, New York, 1991. MR 1105330 (93d:57003)

2.
Benjamin Baumslag.
Residually free groups.
Proc. London Math. Soc. (3), 17:402-418, 1967. MR 0215903 (35:6738)

3.
Gilbert Baumslag.
On generalised free products.
Math. Z., 78:423-438, 1962. MR 0140562 (25:3980)

4.
Gilbert Baumslag.
On a problem of Lyndon.
J. London Math. Soc., 35:30-32, 1960. MR 0111780 (22:2641)

5.
I. Chiswell.
Introduction to $ \Lambda$-trees.
World Scientific, 2001. MR 1851337 (2003e:20029)

6.
I. M. Chiswell.
Some examples of groups with no nontrivial action on a $ \Lambda$-tree.
Mathematika, 42(1):214-219, 1995. MR 1346688 (96g:20036)

7.
Warren Dicks and H. H. Glover.
An algorithm for cellular maps of closed surfaces.
Enseign. Math. (2), 43(3-4):207-252, 1997. MR 1489884 (99e:57001)

8.
A. M. Gaglione and D. Spellman.
Generalizations of free groups: Some questions.
Comm. Algebra, 22(8):3159-3169, 1994. MR 1272379 (95c:20035)

9.
S. M. Gersten and H. B. Short.
Small cancellation theory and automatic groups.
Invent. Math., 102(2):305-334, 1990. MR 1074477 (92c:20058)

10.
V. Guirardel.
Limit groups and groups acting freely on $ \mathbb{R}^n$-trees.
Geometry & Topology, 8:1427-1490, 2004. MR 2119301 (2005m:20060)

11.
T. Hsu and D. Wise.
Cubulating graphs of free groups with cyclic edge groups.
In preparation, 2006.

12.
O. Kharlampovich and A.  Myasnikov.
Irreducible affine varieties over a free group I. Irreducibility of quadratic equations and Nullstellensatz.
Journal of Algebra 200:517-570, 1998. MR 1610664 (2000b:20032b)

13.
R. C. Lyndon.
The equation $ a\sp{2}b\sp{2}=c\sp{2}$ in free groups.
Michigan Math. J, 6:89-95, 1959. MR 0103218 (21:1999)

14.
R. C. Lyndon and M. P. Schützenberger.
The equation $ a\sp{M}=b\sp{N}c\sp{P}$ in a free group.
Michigan Math. J., 9:289-298, 1962. MR 0162838 (29:142)

15.
A. Martino and S. O Rourke.
Free actions on $ \mathbb{Z}\sp n$-trees: A survey.
In Geometric methods in group theory, volume 372 of Contemp. Math., pages 11-25. Amer. Math. Soc., Providence, RI, 2005. MR 2139673 (2006g:20037)

16.

A. Martino and S. O Rourke.
Some free actions on non-Archimedean trees.
Journal of Group Theory 7:275-286, 2004. MR 2049022 (2005b:20047)

17.
V. N. Remeslennikov.
$ \exists$-free groups.
Sibirsk. Mat. Zh., 30(6):193-197, 1989. MR 1043446 (91f:03077)

18.
Eugene Schenkman.
The equation $ a\sp{n}b\sp{n}=c\sp{n}$ in a free group.
Ann. of Math. (2), 70:562-564, 1959. MR 0104723 (21:3476)

19.
Marcel-Paul Schützenberger.
Sur l'équation $ a\sp{2+n}=b\sp{2+m}c\sp{2+p}$ dans un groupe libre.
C. R. Acad. Sci. Paris, 248:2435-2436, 1959. MR 0103219 (21:2000)

20.
Zlil Sela.
Diophantine geometry over groups. I. Makanin-Razborov diagrams.
Publ. Math. Inst. Hautes Études Sci., (93):31-105, 2001. MR 1863735 (2002h:20061)

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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
Email: laura.ciobanu@unifr.ch

A. Martino
Affiliation: Department of Mathematics, Universitat Politècnica de Catalunya, 08860 Castelldefels, Spain
Email: Armando.Martino@upc.edu

S. O Rourke
Affiliation: Department of Mathematics, Cork Institute of Technology, Cork, Ireland
Email: Shane.ORourke@cit.ie

DOI: 10.1090/S0002-9947-08-04639-4
PII: S 0002-9947(08)04639-4
Keywords: Free actions, $\Lambda $-trees, hyperbolic groups, CAT($-1$).
Received by editor(s): December 6, 2006
Posted: August 19, 2008
Copyright of article: Copyright 2008, American Mathematical Society

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