AI 中文总结
本文研究单局部有限群的混合恒等式，推导李型有限单群混合恒等式长度下界，分类容许混合恒等式的无限单局部有限群，还证明单紧李群不容许混合恒等式。
AI 中文摘要
群的混合恒等式是带有常数的非平凡字，在其变量的所有代换下均消失。我们推导李型有限单群中混合恒等式长度的下界，并精确刻画有界秩的这类群族中哪些满足有界长度的混合恒等式。我们对所有容许混合恒等式的无限单局部有限群进行分类：除了明确列出的交错群、有限线性经典群以及非单连通李型群外，不存在这类群。对于该列表中的群，我们确定混合恒等式何时必为奇异的，并得到其临界常数的限制。此外，我们证明单紧李群不容许混合恒等式。
英文摘要
A mixed identity of a group is a nontrivial word with constants that vanishes under every substitution of its variables. We derive lower bounds for the length of mixed identities in finite simple groups of Lie type, and characterise exactly those families of such groups of bounded rank which satisfy mixed identities of bounded length. We classify the infinite simple locally finite groups admitting a mixed identity: apart from an explicit list of alternating, finitary linear classical, and non-simply-laced groups of Lie type, no such group exists. For the groups in this list, we determine when mixed identities must be singular and obtain restrictions on their critical constants. Moreover, we prove that simple compact Lie groups do not admit mixed identities.
Comments36 pages, 1 table
