Free Burnside groups of large odd exponent have cost 1

We prove that the free Burnside group $B(m,n)$ with $m\geq 2$ generators and sufficiently large odd exponent $n$ (e.g., $n\geq 1003$ if $m=2$, and $n\geq 665$ if $m\geq 3$), has cost $1$. It follows that $B(m,n)$ is anti-treeable, and that its first $\ell^2$-Betti number $β_1^{(2)}(B(m,n))$ vanishes. The latter recovers and extends a result of Feldkamp and Kionke [Proc. Amer. Math. Soc., 2023], who proved that $β_1^{(2)}(B(m,p))=0$ for all sufficiently large prime exponents $p$.