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ON THE DYNAMICS OF ENDOMORPHISMS OF THE DIRECT PRODUCT OF TWO FREE GROUPS
Part of:
Topological dynamics
Structure and classification of infinite or finite groups
Special aspects of infinite or finite groups
Abelian groups
Published online by Cambridge University Press: 17 October 2025
Abstract
We prove that Brinkmann’s problems are decidable for endomorphisms of 
$F_n\times F_m$: given 
$(x,y),(z,w)\in F_n\times F_m$ and 
$\Phi \in \mathrm {End}(F_n\times F_m)$, it is decidable whether there is some 
$k\in \mathbb {N}$ such that 
$(x,y)\Phi ^k=(z,w)$ (or 
$(x,y)\Phi ^k\sim (z,w)$). We also prove decidability of a two-sided version of Brinkmann’s conjugacy problem for injective endomorphisms which, from the work of Logan, yields a solution to the conjugacy problem in ascending HNN-extensions of 
$F_n\times F_m$. Finally, we study the dynamics of automorphisms of 
$F_n\times F_m$ at the infinity, proving that that their dynamics at the infinity is asymptotically periodic, as occurs in the free and free-abelian times free cases.
Keywords
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- Copyright
- © The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
This work is funded by national funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020 (Center for Mathematics and Applications).
Communicated by Benjamin Martin
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