Monday, February 23rd, 13:00, CRM Small Room
Speaker: Borja de Balle Pigem, UPC
Title: Extensions de l'algorisme clàssic de Whitehead.
Thursday
Saturday
Seminar February 16th
Monday, February 16th, 13:00, CRM Small Room
Speaker: Xaro Soler Escrivà, Universitat d'Alacant
Title: Algunes qüestions de permutabilitat en grups finits.
Speaker: Xaro Soler Escrivà, Universitat d'Alacant
Title: Algunes qüestions de permutabilitat en grups finits.
Monday
Seminar February 9th
Monday, February 9th, 13:00, CRM Small Room
Speaker: Pedro V. Silva, University of Porto
Title: The mixed orbit problem for the free group of rank 2
Abstract: In 1936, Whitehead solved the orbit problem for words of a free group: given $u,v \in F$, it is decidable whether or not $v =\varphi(u)$ for some $\varphi \in \mbox{Aut} F$. In 1984, Gersten proved a similar theorem for f.g. subgroups of a free group. In both cases, the proofs rely heavily on the use of Whitehead automorphisms to compute shortest words or "smallest" subgroups in the orbit. Such techniques cannot be applied to the mixed orbit problem: given $u \in F$ and $H \leq_{f.g.} F$, is it decidable whether or not $u \in \varphi(H)$ for some $\varphi \in \mbox{Aut} F$? We solve this problem for free groups of rank 2 for the full automorphic orbit and for various subgroups of $\mbox{Aut} F$.
Our techniques involve an appropriate decomposition of $\mbox{Aut} F$ and the study of the dynamical behaviour of a Stallings automaton when we apply automorphisms accordingly. Introducing new concepts such as sources, sinks and singularities for such automata, and proving the existence of certain invariants, we show that the dynamics of such a process is eventually expanding. This fact will lead to the desired decidability results.
These results were obtained in collaboration with Pascal Weil (University of Bordeaux).
Speaker: Pedro V. Silva, University of Porto
Title: The mixed orbit problem for the free group of rank 2
Abstract: In 1936, Whitehead solved the orbit problem for words of a free group: given $u,v \in F$, it is decidable whether or not $v =\varphi(u)$ for some $\varphi \in \mbox{Aut} F$. In 1984, Gersten proved a similar theorem for f.g. subgroups of a free group. In both cases, the proofs rely heavily on the use of Whitehead automorphisms to compute shortest words or "smallest" subgroups in the orbit. Such techniques cannot be applied to the mixed orbit problem: given $u \in F$ and $H \leq_{f.g.} F$, is it decidable whether or not $u \in \varphi(H)$ for some $\varphi \in \mbox{Aut} F$? We solve this problem for free groups of rank 2 for the full automorphic orbit and for various subgroups of $\mbox{Aut} F$.
Our techniques involve an appropriate decomposition of $\mbox{Aut} F$ and the study of the dynamical behaviour of a Stallings automaton when we apply automorphisms accordingly. Introducing new concepts such as sources, sinks and singularities for such automata, and proving the existence of certain invariants, we show that the dynamics of such a process is eventually expanding. This fact will lead to the desired decidability results.
These results were obtained in collaboration with Pascal Weil (University of Bordeaux).
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