Bardakov, Valeriy G.; Gongopadhyay, Krishnendu Palindromic width of free nilpotent groups. (English) Zbl 1308.20034 J. Algebra 402, 379-391 (2014). The notion of palindromic width of a group originates from the combinatorial group theory and has been studied by the first author during the last years. Here the authors consider finitely generated free nilpotent groups and prove that their palindromic width is finite (see Theorem 1.1). They also prove that the same is true for (free abelian)-by-nilpotent groups of finite rank (see Proposition 3.7). In particular, the palindromic width of a finitely generated free metabelian group is finite (see Corollary 3.8). Two interesting open problems remain (see Problems 1 and 2 at pages 380 and 381). The first is of algorithmic nature. The second correlates the palindromic width to the notion of commutator width, which is more usual in the commutator calculus of groups. The conjecture is that the finiteness of the palindromic width is equivalent to that of the commutator width for a finitely generated group. Counterexamples, in fact, are not known at the moment. Reviewer: Francesco G. Russo (Rondebosch) Cited in 1 ReviewCited in 7 Documents MSC: 20F18 Nilpotent groups 20F05 Generators, relations, and presentations of groups 20E05 Free nonabelian groups Keywords:free nilpotent groups; palindromic widths; commutator widths; commutators; finitely generated groups; products of palindromes × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Rhemtulla, A. H., A problem of bounded expressibility in free groups, Proc. Cambridge Philos. Soc., 64, 573-584 (1969) · Zbl 0159.03001 [2] Akhavan-Malayeri, M.; Rhemtulla, A., Commutator length of abelian-by-nilpotent groups, Glasg. Math. J., 40, 1, 117-121 (1998) · Zbl 0911.20028 [3] Allambergenov, Kh. S.; Romanʼkov, V. A., Products of commutators in groups, Dokl. Akad. Nauk UzSSR, 4, 14-15 (1984), (in Russian) · Zbl 0578.20025 [4] Allambergenov, Kh. S.; Romanʼkov, V. A., On products of commutators in groups (1985), 20 pp. (in Russian) [5] Bardakov, V., Computation of commutator length in free groups, Algebra Logic. Algebra Logic, Algebra Logic, 39, 4, 224-251 (2000), (in Russian); translation in · Zbl 0960.20019 [6] Bardakov, V.; Shpilrain, V.; Tolstykh, V., On the palindromic and primitive widths of a free group, J. Algebra, 285, 574-585 (2005) · Zbl 1085.20011 [7] Bardakov, V.; Tolstykh, V., The palindromic width of a free product of groups, J. Aust. Math. Soc., 81, 2, 199-208 (2006) · Zbl 1114.20012 [8] Collins, D., Palindromic automorphism of free groups, (Combinatorial and Geometric Group Theory. Combinatorial and Geometric Group Theory, London Math. Soc. Lecture Note Ser., vol. 204 (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 63-72 · Zbl 0843.20022 [9] Glover, H.; Jensen, C., Geometry for palindromic automorphisms of free groups, Comment. Math. Helv., 75, 644-667 (2000) · Zbl 0972.20021 [10] Gilman, J.; Keen, L., Enumerating palindromes and primitives in rank two free groups, J. Algebra, 332, 1-13 (2011) · Zbl 1237.20023 [11] Gilman, J.; Keen, L., Discreteness criteria and the hyperbolic geometry of palindromes, Conform. Geom. Dyn., 13, 76-90 (2009) · Zbl 1193.30055 [12] Helling, H., A note on the automorphism group of the rank two free group, J. Algebra, 223, 610-614 (2000) · Zbl 0951.20025 [13] Piggott, A., Palindromic primitives and palindromic bases in the free group of rank two, J. Algebra, 304, 359-366 (2006) · Zbl 1111.20028 [14] Magnus, W.; Karrass, A.; Solitar, D., Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations (1996), Interscience Publishers: Interscience Publishers New York This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.