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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.

MSC:

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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