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







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    Thompson’s groups T and V are finitely presented infinite simple groups with Serre’s property FA; in particular, they don’t split.
    – HJRW
    3 hours ago














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







share|cite|improve this question

















  • 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
    3 hours ago












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







share|cite|improve this question













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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










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
    3 hours ago












  • 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
    3 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
3 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
3 hours ago










1 Answer
1






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5
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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.






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  • I see. Thanks for the explanation.
    – Bruno
    1 hour ago










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1 Answer
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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.






share|cite|improve this answer






















  • I see. Thanks for the explanation.
    – Bruno
    1 hour ago














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.






share|cite|improve this answer






















  • I see. Thanks for the explanation.
    – Bruno
    1 hour ago












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.






share|cite|improve this answer














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.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 2 hours ago

























answered 3 hours ago









HJRW

17.9k250115




17.9k250115











  • 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




I see. Thanks for the explanation.
– Bruno
1 hour ago

















 

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