Abstract
We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which—as it happens in the free group—is computable in the finitely generated case.
This approach provides a neat geometric description of (even non-(finitely generated)) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric solution to the subgroup intersection problem and the finite index problem, providing recursive bases and transversals, respectively.
Funding Statement
The first author was partially supported by CMUP (UID/MAT/00144/2019), which is funded by FCT (Portugal) with national (MCTES) and European structural funds through the FEDER programs, under the partnership agreement PT2020. Parts of this project were developed during the participation of the first author in the “Logic and Algorithms in Group Theory” meeting held in the Hausdorff Research Institute for Mathematics (Bonn) in fall 2018. He was also partially supported by MINECO grant PID2019-107444GA-I00 and the Basque Government grant IT974-16.
Both authors acknowledge partial support from the Spanish Agencia Estatal de Investigación, through grant MTM2017-82740-P (AEI/FEDER, UE), and also from the Barcelona Graduate School of Mathematics through the “María de Maeztu” Program for Units of Excellence in R&D (MDM-2014-0445).
Citation
Jordi Delgado. Enric Ventura. "Stallings automata for free-times-abelian groups: intersections and index." Publ. Mat. 66 (2) 789 - 830, 2022. https://doi.org/10.5565/PUBLMAT6622209
Information
