Geodesic
Optimal linear sofic approximations of countable groups
Glasgow mathematical journal, Tome 67 (2025) no. 2, pp. 245-268

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DOI

Let $G$ be a group. The notion of linear sofic approximations of $G$ over an arbitrary field $F$ was introduced and systematically studied by Arzhantseva and Păunescu [3]. Inspired by one of the results of [3], we introduce and study the invariant $\kappa _F(G)$ that captures the quality of linear sofic approximations of $G$ over $F$. In this work, we show that when $F$ has characteristic zero and $G$ is linear sofic over $F$, then $\kappa _F(G)$ takes values in the interval $[1/2,1]$ and $1/2$ cannot be replaced by any larger value. Further, we show that under the same conditions, $\kappa _F(G)=1$ when $G$ is torsion-free. These results answer a question posed by Arzhantseva and Păunescu [3] for fields of characteristic zero. One of the new ingredients of our proofs is an effective non-concentration estimates for random walks on finitely generated abelian groups, which may be of independent interest.
DOI : 10.1017/S0017089524000351
Mots-clés : linear sofic groups, metric approximations, sofic groups 
Mallahi-Karai, Keivan; Yekta, Maryam Mohammadi. Optimal linear sofic approximations of countable groups. Glasgow mathematical journal, Tome 67 (2025) no. 2, pp. 245-268. doi: 10.1017/S0017089524000351
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     title = {Optimal linear sofic approximations of countable groups},
     journal = {Glasgow mathematical journal},
     pages = {245--268},
     year = {2025},
     volume = {67},
     number = {2},
     doi = {10.1017/S0017089524000351},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089524000351/}
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