finitely presented simple group (or more general with trivial profinite completion) that is not amalgamated free product
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As is describled in the title. Is there a finitely presented group $G$, with trivial profinite completion $widehatG=0$, which is not amalgamated free product?
For example, the famous example Higman groups are all amalgamated free product.
gr.group-theory geometric-group-theory
asked 4 hours ago
Bruno
1277
add a comment |Â
up vote
3
down vote
favorite
As is describled in the title. Is there a finitely presented group $G$, with trivial profinite completion $widehatG=0$, which is not amalgamated free product?
For example, the famous example Higman groups are all amalgamated free product.
gr.group-theory geometric-group-theory
asked 4 hours ago
Bruno
1277
5
Thompson’s groups T and V are finitely presented infinite simple groups with Serre’s property FA; in particular, they don’t split.
– HJRW
4 hours ago
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
As is describled in the title. Is there a finitely presented group $G$, with trivial profinite completion $widehatG=0$, which is not amalgamated free product?
For example, the famous example Higman groups are all amalgamated free product.
gr.group-theory geometric-group-theory
asked 4 hours ago
Bruno
1277
As is describled in the title. Is there a finitely presented group $G$, with trivial profinite completion $widehatG=0$, which is not amalgamated free product?
For example, the famous example Higman groups are all amalgamated free product.
gr.group-theory geometric-group-theory
gr.group-theory geometric-group-theory
asked 4 hours ago
Bruno
1277
asked 4 hours ago
Bruno
1277
asked 4 hours ago
Bruno
1277
asked 4 hours ago
Bruno
1277
asked 4 hours ago
Bruno
1277
1277
5
Thompson’s groups T and V are finitely presented infinite simple groups with Serre’s property FA; in particular, they don’t split.
– HJRW
4 hours ago
add a comment |Â
5
Thompson’s groups T and V are finitely presented infinite simple groups with Serre’s property FA; in particular, they don’t split.
– HJRW
4 hours ago
5
5
Thompson’s groups T and V are finitely presented infinite simple groups with Serre’s property FA; in particular, they don’t split.
– HJRW
4 hours ago
Thompson’s groups T and V are finitely presented infinite simple groups with Serre’s property FA; in particular, they don’t split.
– HJRW
4 hours ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
5
down vote
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I'm going to flesh out my comment above to an answer. Thompson's groups $T$ and $V$ are famous examples of finitely presented infinite simple groups.
In this paper of Dan Farley, it is shown that $T$ and $V$ have Serre's property FA, which means that every time they act on a tree there is a global fixed point. (Farley says that this statement is originally due to Ken Brown.) When applied to the Bass--Serre tree of a splitting, it follows that any such splitting of $T$ or $V$ is trivial.
answered 3 hours ago
HJRW
17.9k250115
I see. Thanks for the explanation.
– Bruno
1 hour ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
accepted
I'm going to flesh out my comment above to an answer. Thompson's groups $T$ and $V$ are famous examples of finitely presented infinite simple groups.
In this paper of Dan Farley, it is shown that $T$ and $V$ have Serre's property FA, which means that every time they act on a tree there is a global fixed point. (Farley says that this statement is originally due to Ken Brown.) When applied to the Bass--Serre tree of a splitting, it follows that any such splitting of $T$ or $V$ is trivial.
answered 3 hours ago
HJRW
17.9k250115
I see. Thanks for the explanation.
– Bruno
1 hour ago
add a comment |Â
up vote
5
down vote
accepted
I'm going to flesh out my comment above to an answer. Thompson's groups $T$ and $V$ are famous examples of finitely presented infinite simple groups.
In this paper of Dan Farley, it is shown that $T$ and $V$ have Serre's property FA, which means that every time they act on a tree there is a global fixed point. (Farley says that this statement is originally due to Ken Brown.) When applied to the Bass--Serre tree of a splitting, it follows that any such splitting of $T$ or $V$ is trivial.
answered 3 hours ago
HJRW
17.9k250115
I see. Thanks for the explanation.
– Bruno
1 hour ago
add a comment |Â
up vote
5
down vote
accepted
up vote
5
down vote
accepted
I'm going to flesh out my comment above to an answer. Thompson's groups $T$ and $V$ are famous examples of finitely presented infinite simple groups.
In this paper of Dan Farley, it is shown that $T$ and $V$ have Serre's property FA, which means that every time they act on a tree there is a global fixed point. (Farley says that this statement is originally due to Ken Brown.) When applied to the Bass--Serre tree of a splitting, it follows that any such splitting of $T$ or $V$ is trivial.
answered 3 hours ago
HJRW
17.9k250115
I'm going to flesh out my comment above to an answer. Thompson's groups $T$ and $V$ are famous examples of finitely presented infinite simple groups.
In this paper of Dan Farley, it is shown that $T$ and $V$ have Serre's property FA, which means that every time they act on a tree there is a global fixed point. (Farley says that this statement is originally due to Ken Brown.) When applied to the Bass--Serre tree of a splitting, it follows that any such splitting of $T$ or $V$ is trivial.
answered 3 hours ago
HJRW
17.9k250115
edited 3 hours ago
answered 3 hours ago
HJRW
17.9k250115
answered 3 hours ago
HJRW
17.9k250115
answered 3 hours ago
HJRW
17.9k250115
17.9k250115
I see. Thanks for the explanation.
– Bruno
1 hour ago
add a comment |Â
I see. Thanks for the explanation.
– Bruno
1 hour ago
I see. Thanks for the explanation.
– Bruno
1 hour ago
I see. Thanks for the explanation.
– Bruno
1 hour ago
add a comment |Â
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5
Thompson’s groups T and V are finitely presented infinite simple groups with Serre’s property FA; in particular, they don’t split.
– HJRW
4 hours ago