Strong boundedness and algebraically closed groups. (English) Zbl 1174.20011

A group \(G\) is called strongly bounded if it is Cayley bounded (for every generating set \(U\subset G\) there exists \(n\in\omega\) such that every element of \(G\) is a product of \(n\) elements of \(U\cup U^{-1}\cup\{1\}\)) and cannot be presented as the union of a strictly increasing chain \(\{H_n:n\in\omega\}\) of proper subgroups. A group \(G\) is \(\omega_1\)-existentially closed if for every set \(\sum(\overline x)\) of equalities and inequalities of the form \(w(\overline x,\overline a)=(\neq )1\) depending on variables \(\overline x\) and at most countably many parameters from \(G\), if \(\sum(\overline x)\) has a solution in some group extending \(G\), then it is satisfied already in \(G\). In the paper a new proof of the assertion that every \(\omega_1\)-existentially closed group \(G\) is strongly bounded is presented.


20F05 Generators, relations, and presentations of groups
20F65 Geometric group theory
20A15 Applications of logic to group theory
20E08 Groups acting on trees
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