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Let $H_{1}$ and $H_{2}$ are two handlebodies. If $\partial H_{1}$ and $\partial H_{2}$ are homeomorphic, then $H_{1}\cup_{f} H_{2}$ is a Heegaard splitting. To a general Heegaard splitting, one of $H_{1}$ and $H_{2}$ is allowed to be a compression body. Then the maximal genus boundary of $H_{1}$ is denoted by $\partial _{+}H_{1}$. Similar to $\partial_{+}H_{2}$. If $\partial _{+} H_{1}$ and $\partial_{+}H_{2}$ are homeomorphic, then $H_{1}\cup_{\partial_{+}H_{1}}H_{2}$ is a general Heegaard splitting.

The mapping class group of $H_{1}\cup_{\partial_{+}H_{1}}H_{2}$ is the isotopy class of homeomorphisms of $\partial _{+}H_{1}$, which can be extended to be a homemorphism of $H_{1}$ and $H_{2}$,.

If both $H_{1}$ and $H_{2}$ are two handlebodies, it is the result of Namazi "Big Heegaard distance implies finite mapping class group"that if the Heegaard distance (introduced by Hempel) is large enough then the mapping class group of it is finite. Later Johnson "Mapping class groups of medium distance Heegaard splittings"extended this result and proved that if the Heegaard distance of it is at least 4, the mapping class group of it is finite.

For a general Heegaard splitting, may I ask the following question?

Question: If the Heegaard distance of it is large enough, is the mapping class group of it finitely generated?

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  • $\begingroup$ Perhaps, I am misreading the question, but given the Johnson result, can your question be rephrased as "Let $H_1 \cup_f H_2$ be a Heegaard splitting of distance 1, 2, or 3. Is the mapping class group of $H_1 \cup_f H_2$ finitely generated?" $\endgroup$ Jan 20, 2017 at 16:00

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