Let $H$ be a finitely generated subgroup of the free group $F(A)$ and $G_P$ the pseudovariety of all finite $p$-group with $p$ fixed prime number. We endow $F(A)$ with the pro-$G_p$ topology. Suppose that $H$ is $p$-dense. What can we say about the $p$-kernel of the transition monoid of $H$?
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$\begingroup$ You should give the definitions you are using. By the transition monoid of $H$, do you mean the finite inverse monoid associated with the DFA accepting $H$? Is the $p$-kernel of a finite monoid $M$ the intersection of all sets $f^{-1}(1)$ where $f$ is a relational morphism from $M$ into a finite $p$-group? And what do you mean by $H$ is $p$-dense? $\endgroup$– J.-E. PinCommented Apr 6, 2015 at 15:17
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$\begingroup$ You are right about the definition of transition monoid and $p$-kernel. By $p$-dense I mean the $p$-closure of the subgroup $H$ is $F(A)$. $\endgroup$– user182085Commented Apr 6, 2015 at 15:32
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$\begingroup$ If we denote by $M(H)$ the transition monoid of $H$ and $K_{G_p}(M(H))$ be the p-kernel of $M(H)$, I want to know when the equality $M(H)=K_{G_p}(M(H))$ holds. Is it true if $H$ is p-dense? $\endgroup$– user182085Commented Apr 6, 2015 at 15:41
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$\begingroup$ Being p-dense is equivalent to the p-kernel acting transitively on the Stallings graph. I don't immediately see why this forces the p-kernel to be everything $\endgroup$– Benjamin SteinbergCommented Apr 6, 2015 at 16:00
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