Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 104 / 5

Letbe the group of finitely supported functions $f:G\to H$ with pointwise multiplication. The left-translation actiondefines the restricted wreath product

Suppose $X$ and $Y$ are finite generating sets for $G$ and $H$. Embed each $y\in Y$ as a lamp supported at the identity of $G$. Conjugating these lamps by words in $X$ produces copies of $Y$ at every coordinate, and these copies generate $B$. Thus $X$ together with the identity-coordinate copy of $Y$ is a finite generating set for $H\wr G$.

Let $\theta :H\wr G\to Q$ be any homomorphism to a finite group. Because $G$ is infinite, two distinct elements $r,s\in G$ have $\theta (r)=\theta (s)$. For $h\in H$, denote by $h_g$ the lamp with value $h$ at $g$. Conjugation translates lamps, sofor every $h\in H$. Choose $h,k\in H$ with $[h,k]\ne 1$. Lamps at different coordinates commute, and thereforeBut $[h_r,k_r]$ is the nonidentity lamp $[h,k{]}_r$. This same nonidentity element is killed by every finite quotient, so $H\wr G$ is not residually finite.

Choose $1\ne h\in H$ and let $t$ generate $\mathbb{Z}$. For each binary string $({\epsilon }_0,\dots ,{\epsilon }_{n-1})$, the wordrecords that string in the lamps at positions $0,1,\dots ,n-1$. The resulting $2^n$ group elements are distinct and have word length at most $3n$ with respect to any finite generating set containing $h$ and $t$. Hence the growth function is bounded below exponentially, and $H\wr \mathbb{Z}$ has exponential growth.

Write an element of $H\wr \mathbb{Z}$ as $(f,t^n)$. If it is central, commuting with $t$ makes the finitely supported lamp configuration $f$ invariant under translation. The only such configuration on the infinite set $\mathbb{Z}$ is the identity, so $f=1$. If $n\ne 0$, then $t^n$ moves a nonidentity lamp at position zero to position $n$ and does not commute with it. Therefore $n=0$, and