Notes of Elisabeth Fink’s lecture

Width of simple and wreath product groups with respect to palindromes

With Andreas Thom.

1. Width

Let {G} be a group, {X\subset G} a set, possibly infinite. Say {G} has width {N} with respect to {X} if every element is the product of at most {N} elements of {X}.

Examples.

  1. {G} infinite, {X} finite. Then width is infinite.
  2. {X=\{} commutators {\}}, {G^1=[G,G]}. Commutator width is 1 for finite simple groups.
  3. Bi-invariant metrics. Let {X} be the smallest subset of {G} containing all generators and closed under conjugation. For instance, diffeomorphism groups, symplectomorphism groups have biinvariant metrics, and so infinite width with respect to {X}. I have studied the case of Grigorchuk group.

2. Palindromes

Here, I will consider the set of palindromes.

Let {F} be a free group. An element {w\in F} is a \textrm{palindrome} if it is reduced and reads the same in both directions.

Let {G} be a finitely generated group, {G=F/N}. An element of {G} is a palindrom if one of its inverse images in {F} is.

Example. {G=<a,b,c|abc>}. Then {abba}, {ababcba} are palindromes.

Notation. {\bar{g}=} the reverse element of some presentation of {g}.

Examples of groups with finite palindromic width.

  1. Free metabelian groups (Bardakov-Gongopadhay, Riley-Sale).
  2. certain solvable-by-nilpotent groups.

Examples of groups with infinite palindromic width.

  1. Free groups (Bardakov-?).
  2. Groups with free quotients.

3. Relations in groups

Proposition 1 Let {G} be a non abelian group with no free quotient, generated by {S}. Let {\tilde{S}} be obtained from {S} with one Nielsen transformation, and {\tilde{R}} the adapted set of relators. Then there exists {r\in\tilde{R}} such that {\bar{r}\not=1}, {r=1} in {G}.

Proposition 2 The set {X=\{r\bar{r}\,;\,r=1\}} is a normal subgroup of {G}.

Theorem 3 (Fink-Thom) Let {G} be a simple group generated by finite set {S}. Then, with respect to {\tilde{S}}, every element is a palindrome.

Indeed, the set {X} is a non trivial normal subgroup, so {X=G}.

Theorem 4 Let {G} be a just infinite group generated by finite set {S}. Then, with respect to {\tilde{S}}, the palindromic width is finite.

4. Wreath products

A permutation wreath product is a semi-direct product {G\wr H=(\prod_{H}G)\times H}.

Theorem 5 Let {H} be a non abelian group. Assume that {H} has palindromic width {m} with respect to some subset {X}. Then any element of {F_r\wr H} is a product of at most {m+1} palindromes with respect to generating set {(S,1)\cup(1,\tilde{X})}, where {F_r} has generating set {S}.

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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