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misread original question and qualified a comment
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NWMT
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Grushko's Theorem, which is a predecessor of Swarup's result, states $\mathrm{rank}(A*B) = \mathrm{rank}(A)+\mathrm{rank}(B)$, where $A*B$ denotes the free product and rank denotes the cardinality of a minimal generating set.

This means that an infinite rank subgroup can never be a free factor of a finite rank free group. For this reason there should be no result analogous to Swarup's forThe situation with infinite rank edge and vertex groups.

That said, you could try to find some generalization of "free factor" that would be useful, but I think this is unlikelyqualitatively very different. This is becauseFor example, the class finite rank groups that arise as HNN extensions of infinite rank free groups is surprisingly rich! The following paper by Hagen and Wise shows that a non-trivial class of hyperbolic groups (constructible groups) are virtually HNN extensions of (infinite rank) free groups Special groups with an elementary hierarchy are virtually free-by-ℤ.

All this to say that the infinite rank edge group case is qualitatively quite different from the finite rank edge group case.

On a technical note, in Swarup's original paper, the paper by Diao and Feighn mentionned in another answer, and my own paper On the one-endedness of graphs of groups which is inspired by Diao-Feighn, finite generation of the edge groups plays a critical role. In the more modern papers, finite generation of edge groups ensures that the group acts co-compactly on a nice square complex. Therefore, if these methods can be generalized

All this to the infinite rank edge group case, there will needsay that a generalization of Swarup's Theorem to your situation may require to be some new ideas.

Grushko's Theorem, which is a predecessor of Swarup's result, states $\mathrm{rank}(A*B) = \mathrm{rank}(A)+\mathrm{rank}(B)$, where $A*B$ denotes the free product and rank denotes the cardinality of a minimal generating set.

This means that an infinite rank subgroup can never be a free factor of a finite rank free group. For this reason there should be no result analogous to Swarup's for infinite rank edge groups.

That said, you could try to find some generalization of "free factor" that would be useful, but I think this is unlikely. This is because the class finite rank groups that arise as HNN extensions of infinite rank free groups is surprisingly rich! The following paper by Hagen and Wise shows that a non-trivial class of hyperbolic groups (constructible groups) are virtually HNN extensions of (infinite rank) free groups Special groups with an elementary hierarchy are virtually free-by-ℤ.

All this to say that the infinite rank edge group case is qualitatively quite different from the finite rank edge group case.

On a technical note, in Swarup's original paper, the paper by Diao and Feighn mentionned in another answer, and my own paper On the one-endedness of graphs of groups which is inspired by Diao-Feighn, finite generation of the edge groups plays a critical role. In the more modern papers, finite generation of edge groups ensures that the group acts co-compactly on a nice square complex. Therefore, if these methods can be generalized to the infinite rank edge group case, there will need to be some new ideas.

The situation with infinite rank edge and vertex groups is qualitatively very different. For example, the following paper by Hagen and Wise shows that a non-trivial class of hyperbolic groups (constructible groups) are virtually HNN extensions of (infinite rank) free groups Special groups with an elementary hierarchy are virtually free-by-ℤ.

On a technical note, in Swarup's original paper, the paper by Diao and Feighn mentionned in another answer, and my own paper On the one-endedness of graphs of groups which is inspired by Diao-Feighn, finite generation of the edge groups plays a critical role. In the more modern papers, finite generation of edge groups ensures that the group acts co-compactly on a nice square complex.

All this to say that a generalization of Swarup's Theorem to your situation may require to be some new ideas.

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NWMT
NWMT
  • 1k
  • 5
  • 11

Grushko's Theorem, which is a predecessor of Swarup's result, states $\mathrm{rank}(A*B) = \mathrm{rank}(A)+\mathrm{rank}(B)$, where $A*B$ denotes the free product and rank denotes the cardinality of a minimal generating set.

This means that an infinite rank subgroup can never be a free factor of a finite rank free group. For this reason there should be no result analogous to Swarup's for infinite rank edge groups.

That said, you could try to find some generalization of "free factor" that would be useful, but I think this is unlikely. This is because the class finite rank groups that arise as HNN extensions of infinite rank free groups is surprisingly rich! The following paper by Hagen and Wise shows that a non-trivial class of hyperbolic groups (constructible groups) are virtually HNN extensions of (infinite rank) free groups Special groups with an elementary hierarchy are virtually free-by-ℤ.

All this to say that the infinite rank edge group case is qualitatively quite different from the finite rank edge group case.

On a technical note, in Swarup's original paper, the paper by Diao and Feighn mentionned in another answer, and my own paper On the one-endedness of graphs of groups which is inspired by Diao-Feighn, finite generation of the edge groups plays a critical role. In the more modern papers, finite generation of edge groups ensures that the group acts co-compactly on a nice square complex. Therefore, if these methods can be generalized to the infinite rank edge group case, there will need to be some new ideas.