Free Burnside groups of large odd exponent have cost 1

Abstract

We prove that for every non-elementary torsion-free hyperbolic group $H$ and sufficiently large odd $n>1$, the quotient $H/H^n$ has cost $1$. In particular, when $H$ is the free group of rank $m\ge 2$, we obtain that the free Burnside group $B(m,n)$ has cost $1$ for every odd exponent $n\ge 665$. It follows that $B(m,n)$ is anti-treeable, and that its first ${\ell }^2$-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 $p$.

arXiv commentWe write the result in the more general framework of periodic quotients of non-elementary torsion-free hyperbolic groups, obtaining the free Burnside result as a particular case