Geodesic
First-order model theory and Kaplansky’s stable finiteness conjecture for surjunctive groups
Tullio Ceccherini-Silberstein1 orcid ; Michel Coornaert2 ; Xuan Kien Phung3 orcid
1Università del Sannio, Benevento, Italy; Istituto Nazionale di Alta Matematica “Francesco Severi”, Roma, Italy
2Université de Strasbourg, France
3Université de Montréal, Montréal, Québec, Canada
Groups, geometry, and dynamics, Tome 19 (2025) no. 2, pp. 495-503

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DOI

Using algebraic geometry methods, the third author proved that the group ring of a surjunctive group with coefficients in a field is always stably finite. In other words, every group satisfying Gottschalk’s conjecture also satisfies Kaplansky’s stable finiteness conjecture. Here, we present a proof of this result based on the first-order model theory.
DOI : 10.4171/ggd/885
Classification : 37B15, 03C98, 16S34, 20C07, 68Q80
Mots-clés : surjunctive group, group ring, stably finite ring, model theory, Gottschalk’s conjecture, Kaplansky’s stable finiteness conjecture

Tullio Ceccherini-Silberstein  1   ; Michel Coornaert  2   ; Xuan Kien Phung  3

1 Università del Sannio, Benevento, Italy; Istituto Nazionale di Alta Matematica “Francesco Severi”, Roma, Italy
2 Université de Strasbourg, France
3 Université de Montréal, Montréal, Québec, Canada 
Tullio Ceccherini-Silberstein; Michel Coornaert; Xuan Kien Phung. First-order model theory and Kaplansky’s stable finiteness conjecture for surjunctive groups. Groups, geometry, and dynamics, Tome 19 (2025) no. 2, pp. 495-503. doi: 10.4171/ggd/885
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     author = {Tullio Ceccherini-Silberstein and Michel Coornaert and Xuan Kien Phung},
     title = {First-order model theory and {Kaplansky{\textquoteright}s} stable finiteness conjecture for surjunctive groups},
     journal = {Groups, geometry, and dynamics},
     pages = {495--503},
     year = {2025},
     volume = {19},
     number = {2},
     doi = {10.4171/ggd/885},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/885/}
}
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