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In Gersten's "Problems on Automatic Groups", Problem 14, he asks the following question: Let $G=A\ast_{C}B$ where $A$ and $B$ are automatic and $C$ is infinite cyclic. Is $G$ automatic?

Is this question still open? Or any progress has been made? I am no expert in automatic groups, any reference will be very helpful!

Additional Question: Do people know how to compute the Dehn function of G?

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    $\begingroup$ I believe this is still open, certainly in full generality. It has been proved in various specific cases - for example, if $A$ and $B$ are hyperbolic relative to collections of abelian subgroups, one of which is $C$. $\endgroup$
    – Derek Holt
    Jul 4, 2018 at 10:04
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    $\begingroup$ It is perhaps worth noticing that BS(1,2) gives a negative answer if you replace amalgamated free product by HNN extension. $\endgroup$
    – NWMT
    Jul 5, 2018 at 13:31
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    $\begingroup$ @NWMT Thanks. I remember I have seen this result but I can't remember where it is now. $\endgroup$
    – YCC
    Jul 5, 2018 at 19:44

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