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On the isolated points in the space of groups. (English) Zbl 1132.20018

The authors study isolated points in the space \(\mathcal G_m\) of marked groups of rank \(m\). Examples of such groups are given besides the known ones consisting of finite groups and finitely presented simple groups.
In a group \(G\), a subset \(F\) is a ‘discriminating subset’ if every nontrivial normal subgroup of \(G\) contains an element of \(F\). \(G\) is called ‘finitely discriminable’ if it has a finite discriminating subset.
Proposition 2. A group is isolated if and only if it is finitely presented and finitely discriminable.
The authors study finite discriminable groups and show that an isolated group has solvable word problem.
Then it is shown: Theorem 9. Every finitely generated group is a quotient of an isolated group.
In the end they present examples of isolated groups like Thompson’s group \(F\) and a 3-solvable non-Hopfian group. They also present an infinite isolated group with Kazhdan’s property (T).

MSC:

20F05 Generators, relations, and presentations of groups
20E34 General structure theorems for groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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