## 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. ${G=}$. 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 ${S}$.