Translation equivalent elements in free groups
-
Donghi Lee
Abstract
Let Fn be a free group of rank n ≥ 2. Two elements g, h in Fn are said to be translation equivalent in Fn if the cyclic length of φ(g) equals the cyclic length of φ(h) for every automorphism φ of Fn. Let F(a, b) be the free group generated by {a, b} and let w(a, b) be an arbitrary word in F(a, b). We prove that w(g, h) and w(h, g) are translation equivalent in Fn whenever g, h ∈ Fn are translation equivalent in Fn, and thereby give an affermative solution to problem F38b in the online version (http://www.grouptheory.info) of [G. Baumslag, A. G. Myasnikov and V. Shpilrain. Open problems in combinatorial group theory, 2nd edition. In Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), Contemp. Math. 296 (American Mathematical Society, 2002), pp. 1–38.].
© Walter de Gruyter
Articles in the same Issue
- Crossover Morita equivalences for blocks of the covering groups of the symmetric and alternating groups
- On a theorem of Artin. II
- Cycle index methods for finite groups of orthogonal type in odd characteristic
- Characterization of injectors in finite soluble groups
- Some class size conditions implying solvability of finite groups
- On t-pure and almost pure exact sequences of LCA groups
- Translation equivalent elements in free groups
- On representing words in the automorphism group of the random graph
Articles in the same Issue
- Crossover Morita equivalences for blocks of the covering groups of the symmetric and alternating groups
- On a theorem of Artin. II
- Cycle index methods for finite groups of orthogonal type in odd characteristic
- Characterization of injectors in finite soluble groups
- Some class size conditions implying solvability of finite groups
- On t-pure and almost pure exact sequences of LCA groups
- Translation equivalent elements in free groups
- On representing words in the automorphism group of the random graph
