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Limit groups as limits of free groups. (English) Zbl 1103.20026
By the authors’ definition a marked group $$(G,S)$$ consists of a group $$G$$ with a prescribed tuple $$S=\{s_1,\dots,s_n\}$$ of generators. Two marked groups $$(G_1,S_1)$$, $$(G_2,S_2)$$ are isomorphic as marked groups if and only if the natural bijection of $$S_1$$ onto $$S_2$$ extends to an isomorphism of $$G_1$$ onto $$G_2$$. The set $${\mathcal G }_n$$ of all marked groups with $$n$$ generators is considered as a compact topological space.
It appears that the limits of free groups in $${\mathcal G}_n$$ are Remeslennikov’s finitely generated fully residually free groups, or Sela’s limit groups. The authors’ topological approach to these groups gives some new insight in the relations between fully residually free groups, the universal theory of free groups, ultraproducts and non-standard free groups.

##### MSC:
 20F05 Generators, relations, and presentations of groups 20E05 Free nonabelian groups 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20E26 Residual properties and generalizations; residually finite groups 03C60 Model-theoretic algebra 57M07 Topological methods in group theory 22A05 Structure of general topological groups
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