论文标题
关于不变的准态和稳定混合换向器长度的调查
Survey on invariant quasimorphisms and stable mixed commutator length
论文作者
论文摘要
如果$ ϕ（gxg^{ - 1}）在g $中，每$ g \ gxg^{ - 1}）在普通子组$ n $ of $ g $上的均质准晶体$ ϕ $，则为$ g $ -invariant。不变的准晶体自然出现在符号几何形状和准态性的扩展问题中。此外，众所周知，不可扩展不变的准畸形的存在与稳定的混合换向器长度$ \ mathrm {scl} _ {g，n} $的行为密切相关，这是稳定的稳定的交换器长度$ \ mathrm {scl {scl} _g $。 在这项调查中，我们回顾了不变的准畸形和稳定混合换向器长度的历史和最新发展。我们视为的主题包括几个不变的准畸形的例子，Bavard的二元性定理，用于不变的准畸形，自动不变的准畸形以及对不可延迟的准畸形空间维度的估计。我们还提到了部分准态的扩展问题。
A homogeneous quasimorphism $ϕ$ on a normal subgroup $N$ of $G$ is said to be $G$-invariant if $ϕ(gxg^{-1}) = ϕ(x)$ for every $g \in G$ and for every $x \in N$. Invariant quasimorphisms have naturally appeared in symplectic geometry and the extension problem of quasimorphisms. Moreover, it is known that the existence of non-extendable invariant quasimorphisms is closely related to the behavior of the stable mixed commutator length $\mathrm{scl}_{G,N}$, which is a certain generalization of the stable commutator length $\mathrm{scl}_G$. In this survey, we review the history and recent developments of invariant quasimorphisms and stable mixed commutator length. The topics we treat include several examples of invariant quasimorphisms, Bavard's duality theorem for invariant quasimorphisms, Aut-invariant quasimorphisms, and the estimation of the dimension of spaces of non-extendable quasimorphisms. We also mention the extension problem of partial quasimorphisms.
