A group-word w is called concise if whenever the set of w-values in a group G is finite it always follows that the verbal subgroup w(G) is finite. More generally, a word w is said to be concise in a class of groups X if whenever the set of w-values is finite for a group G in the class X, it always follows that w(G) is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w is a multilinear commutator and q is a prime-power, then the word w^q is indeed concise in the class of residually finite groups. Further, we show that in the case where w=γ_k the word w^q is boundedly concise in the class of residually finite groups. It remains unknown whether the word w^q is actually concise in the class of all groups.

On words that are concise in residually finite groups / Acciarri, C; Shumyatsky, P. - In: JOURNAL OF PURE AND APPLIED ALGEBRA. - ISSN 0022-4049. - 218:1(2014), pp. 130-134. [10.1016/j.jpaa.2013.04.018]

On words that are concise in residually finite groups

Acciarri C;
2014-01-01

Abstract

A group-word w is called concise if whenever the set of w-values in a group G is finite it always follows that the verbal subgroup w(G) is finite. More generally, a word w is said to be concise in a class of groups X if whenever the set of w-values is finite for a group G in the class X, it always follows that w(G) is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w is a multilinear commutator and q is a prime-power, then the word w^q is indeed concise in the class of residually finite groups. Further, we show that in the case where w=γ_k the word w^q is boundedly concise in the class of residually finite groups. It remains unknown whether the word w^q is actually concise in the class of all groups.
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On words that are concise in residually finite groups / Acciarri, C; Shumyatsky, P. - In: JOURNAL OF PURE AND APPLIED ALGEBRA. - ISSN 0022-4049. - 218:1(2014), pp. 130-134. [10.1016/j.jpaa.2013.04.018]
Acciarri, C; Shumyatsky, P
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1255521
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