词映射的同态逼近性
Approximability of word maps by homomorphisms
摘要：
推广了Mann的一个结果，证明了存在一个显函数$f:\left(0,1\right]\rightarrow\left(0,1\right]$，使得对于每个约简字$w$，比如在$d$变量中，至多在$3d$变量中存在一个显约简字$v$（如果$w$的长度至少为$2$，则为非平凡的），使得对于所有$ρ\in\left(0,1\right]$，以下成立：如果$G$是任何有限群，对于该有限群，词映射$w_G:G^d\rightarrow G$在至少$ρ|G|^d$多个参数上与某个固定同态$G^d\rightarrow G$一致，则词映射$v_G:G^{3d}\rightarrow G$具有大小至少为$f(ρ)|G|^{3d}$的纤维。我们还讨论了这个结果的一些应用。
英文摘要：
Generalizing a recent result of Mann, we show that there is an explicit function $f:\left(0,1\right]\rightarrow\left(0,1\right]$ such that for every reduced word $w$, say in $d$ variables, there is an explicit reduced word $v$ in at most $3d$ variables (nontrivial if the length of $w$ is at least $2$) such that for all $ρ\in\left(0,1\right]$, the following holds: If $G$ is any finite group for which the word map $w_G:G^d\rightarrow G$ agrees with some fixed homomorphism $G^d\rightarrow G$ on at least $ρ|G|^d$ many arguments, then the word map $v_G:G^{3d}\rightarrow G$ has a fiber of size at least $f(ρ)|G|^{3d}$. We also discuss some applications of this result.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
