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Optimal linear sofic approximations of countable groups
- Part of
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- Journal:
- Glasgow Mathematical Journal , First View
- Published online by Cambridge University Press:
- 24 January 2025, pp. 1-24
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- Article
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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.
