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- W2593232613 abstract "The emph{normal rank} of a group is the minimal number of elements whose normal closure coincides with the group. We study the relation between the normal rank of a group and its first $ell^2$-Betti number and conjecture that inequality $beta_1^{(2)}(G)$ does not exceed normal rank minus 1 for torsion free groups. The conjecture is proved for limits of left-orderable amenable groups. On the other hand, for every $nge 2$ and every $e>0$, we give an example of a simple group $Q$ (with torsion) such that $beta_1^{(2)}(Q) geq n-1-epsilon$. These groups also provide examples of simple groups of rank exactly $n$ for every $nge 2$; existence of such examples for $n> 3$ was unknown until now." @default.
- W2593232613 created "2017-03-16" @default.
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- W2593232613 date "2012-06-16" @default.
- W2593232613 modified "2023-10-17" @default.
- W2593232613 title "Normal generation and $$ell ^2$$ -Betti numbers of groups" @default.
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- W2593232613 doi "https://doi.org/10.1007/s00208-012-0828-7" @default.
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