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Edited:

The paper Dehn twists have roots proves that Dehn twists have roots, naturally. The limits on that construction can be found in the paper Roots of Dehn twists; they bound the degree of the root (linearly, as I recall) in terms of the complexity of the surface.

Now, if I understand correctly, you are asking for ways to realize $\tau_\alpha^p$, a power of a Dehn twist about $\alpha$, as a product of other twists. If we found a root of $\tau_\alpha^p$, which was not itself a twist, then that would answer your question. But I think that the main idea behind the above papers will work. For: any root of $\tau_\alpha^p$ has the same canonical reduction system, namely $\alpha$. So the root of $\tau_\alpha^p$ stabilizes $\alpha$. You can now build a periodic map in the complement of $\alpha$ that does a fractional twist about $\alpha$: for example $p/q$ fraction of a right twist. Take the $q$-power of this get the desired $\tau_\alpha^p$.

Note that it is far from clear that these constructions are the only way to get a subgroup HH of the type you desire.

Edited:

The paper Dehn twists have roots proves that Dehn twists have roots, naturally. The limits on that construction can be found in the paper Roots of Dehn twists; they bound the degree of the root (linearly, as I recall) in terms of the complexity of the surface.

Now, if I understand correctly, you are asking for ways to realize $\tau_\alpha^p$, a power of a Dehn twist about $\alpha$, as a product of other twists. If we found a root of $\tau_\alpha^p$, which was not itself a twist, then that would answer your question. But I think that the main idea behind the above papers will work. For: any root of $\tau_\alpha^p$ has the same canonical reduction system, namely $\alpha$. So the root of $\tau_\alpha^p$ stabilizes $\alpha$. You can now build a periodic map in the complement of $\alpha$ that does a fractional twist about $\alpha$: for example $p/q$ fraction of a right twist. Take the $q$-power of this get the desired $\tau_\alpha^p$.

Note that it is far from clear that these constructions are the only way to get a subgroup $H$ of the type you desire.

Edited:

The paper Dehn twists have roots proves that Dehn twists have roots, naturally enough. The limits on their construction can be found in the paper Roots of Dehn twists; they bound the degree of the root (linearly, as I recall) in terms of the complexity of the surface.

Now, if I understand correctly, you are asking for ways to realize $\tau_\alpha^k$, a power of a Dehn twist about $\alpha$, as a product of other twists. So, if we found a root of $\tau_\alpha^k$, which was not itself a twist, then that would answer the question. But the main idea behind those papers should be able to do this, as well. That is - any root of $\tau_\alpha^k$ has the same canonical reduction system as that of $\tau_\alpha^k$, namely $\alpha$ itself. So the root stabilizes $\alpha$. You can now build a periodic map in the complement of $\alpha$ that does a fractional twist about $\alpha$: for example three-halves of a right twist. Square this to get $\tau_\alpha^3$.

Note that it is far from clear that these constructions are the only way to get a subgroup $H$ of the type you desire.

Edited:

The paper Dehn twists have roots proves that Dehn twists have roots, naturally. The limits on that construction can be found in the paper Roots of Dehn twists; they bound the degree of the root (linearly, as I recall) in terms of the complexity of the surface.

Now, if I understand correctly, you are asking for ways to realize $\tau_\alpha^p$, a power of a Dehn twist about $\alpha$, as a product of other twists. If we found a root of $\tau_\alpha^p$, which was not itself a twist, then that would answer your question. But I think that the main idea behind the above papers will work. For: any root of $\tau_\alpha^p$ has the same canonical reduction system, namely $\alpha$. So the root of $\tau_\alpha^p$ stabilizes $\alpha$. You can now build a periodic map in the complement of $\alpha$ that does a fractional twist about $\alpha$: for example $p/q$ fraction of a right twist. Take the $q$-power of this get the desired $\tau_\alpha^p$.

Note that it is far from clear that these constructions are the only way to get a subgroup HH of the type you desire.

Examples with $k > 2$ can be found in the paper Dehn twists have roots. The limits on their construction can be found in the paper Roots of Dehn twists; they bound $k$ (linearly, as I recall) in terms of the complexity of the surface. However, it is not clear that these constructions are the only way to get a subgroup $H$ of the type you desire.

Edited:

The paper Dehn twists have roots proves that Dehn twists have roots, naturally enough. The limits on their construction can be found in the paper Roots of Dehn twists; they bound the degree of the root (linearly, as I recall) in terms of the complexity of the surface.

Now, if I understand correctly, you are asking for ways to realize $\tau_\alpha^k$, a power of a Dehn twist about $\alpha$, as a product of other twists. So, if we found a root of $\tau_\alpha^k$, which was not itself a twist, then that would answer the question. But the main idea behind those papers should be able to do this, as well. That is - any root of $\tau_\alpha^k$ has the same canonical reduction system as that of $\tau_\alpha^k$, namely $\alpha$ itself. So the root stabilizes $\alpha$. You can now build a periodic map in the complement of $\alpha$ that does a fractional twist about $\alpha$: for example three-halves of a right twist. Square this to get $\tau_\alpha^3$.

Note that it is far from clear that these constructions are the only way to get a subgroup $H$ of the type you desire.