By Namazi and Johnson’s results, for any distance at least 4 Heegaard splitting, its mapping class group is finite. In contrast, Namazi showed that for a weakly reducible Heegaard splitting, its mapping class group is infinite; Long constructed an irreducible Heegaard splitting where its mapping class group contains a pseudo anosov map. Thus it is interesting to know that for a strongly irreducible but distance at most 3 Heegaard splitting, when its mapping class group is finite.

In [19], Qiu and Zou introduced the definition of a locally large distance 2 Heegaard splitting. Extending their definition into a locally large strongly irreducible Heegaard splitting, we proved that its mapping class group is finite. Moreover, for a toroidal 3-manifold which admits a locally large distance 2 Heegaard splitting in [19], we prove that its mapping class group is finite.

中文翻译：

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映射类组的有限性：局部大的不可约的强烈Heegaard分裂

根据Namazi和Johnson的结果，对于至少4个Heegaard分裂的任何距离，其映射类组都是有限的。相反，纳马齐（Namazi）表明，对于弱可约Heegaard分裂，其映射类组是无限的。Long构造了一个不可约的Heegaard分裂，其映射类组包含一个伪anosov映射。因此，有趣的是，对于一个强不可约但距离最多为3的Heegaard分裂，当其映射类组为有限时。

在[19]中，邱和邹介绍了局部大距离2 Heegaard分裂的定义。将它们的定义扩展到局部的大型不可约Heegaard分裂中，我们证明了它的映射类组是有限的。此外，对于在[19]中允许局部大距离2 Heegaard分裂的环形3形流形，我们证明了它的映射类组是有限的。