Free Burnside groups of large odd exponent have cost 1

We prove that the free Burnside group $B(m,n)$ with $m\ge 2$ generators and sufficiently large odd exponent $n$ (e.g., $n\ge 1003$ if $m=2$, and $n\ge 665$ if $m\ge 3$), has cost $1$. It follows that $B(m,n)$ is anti-treeable, and that its first ${\mathrm{\ell }}^2$-Betti number ${\beta }_1^{(2)}(B(m,n))$ vanishes. The latter recovers and extends a result of Feldkamp and Kionke [Proc. Amer. Math. Soc., 2023], who proved that ${\beta }_1^{(2)}(B(m,p))=0$ for all sufficiently large prime exponents $p$.