发表机构
University of Münster（明斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明了满足特定条件的自由Burnside群具有代价1，进而得到其反树可化性及第一ℓ²-Betti数为零的结论，推广了Feldkamp与Kionke的相关结果。
AI 中文摘要
我们证明，具有m≥2个生成元且指数n为足够大奇数的自由Burnside群B(m,n)（例如，当m=2时n≥1003，当m≥3时n≥665）具有代价1。由此可得，B(m,n)是反树可化的，且其第一ℓ²-Betti数β₁⁽²⁾(B(m,n))为零。这一结果恢复并推广了Feldkamp与Kionke[《美国数学会汇刊》，2023年]的结论，他们证明了对所有足够大的素数指数p，β₁⁽²⁾(B(m,p))=0。
英文摘要
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 $n\geq 665$. 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 $p$.
CommentsWe 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
