AI 中文总结
本文提出左正则表示的重心块分解，证明有限生成群的算子值Haagerup不等式，对两类群得到双边或带特定误差的球块估计，完善相关不等式的下界与估计结果。
AI 中文摘要
我们引入左正则表示的一种基于重心的块分解，并利用它来证明具有重心映射的有限生成群的算子值Haagerup不等式。该分解将三个重心增长条件分离为不同的算子范数贡献，并且在相关Schur乘子的因子化假设下，得到互补的下界。对于秩为ν的中值群，重心块与球算子相关，给出带有显式因子$\binom{\nu+r}{r}$的双边估计；这尤其适用于直角Artin群和作用在有限秩$\text{CAT}(0)$立方复形上的群。对于有限秩粗中值群，我们构造了误差与输入集基数无关的粗迭代中值，并对任意$\text{ε}>0$得到带有因子$o(r^{1+\nu/2+\text{ε}})$的球块估计。
英文摘要
We introduce a centroid-based block decomposition of the left-regular representation and use it to prove operator-valued Haagerup inequalities for finitely generated groups with centroid maps. The decomposition separates the three centroid growth conditions into distinct operator-norm contributions and, under a factorization hypothesis for the associated Schur multipliers, yields complementary lower bounds. For median groups of rank $ν$, the centroid blocks are related to spherical operators, giving two-sided estimates with the explicit factor $\binom{ν+r}{r}$; this applies in particular to right-angled Artin groups and to groups acting on finite-rank $\operatorname{CAT}(0)$ cube complexes. For finite-rank coarse median groups, we construct a coarse iterate median with error independent of the cardinality of the input set and obtain spherical-block estimates with factor $o(r^{1+ν/2+\varepsilon})$ for every $\varepsilon>0$.
