In mathematics, specifically in group theory and logics, limit groups are the finitely generated groups that admit a presentation which is a limit of free group presentations in the discrete Chabauty topology. Formerly known as fully residually free groups, they arise naturally in the study of equations in free groups and have gained significance through the work of Sela on Tarski's problem. They now form a well-studied class of examples in geometric group theory and have led to generalizations such as limit groups over hyperbolic and certain relatively hyperbolic groups. Basic examples include free groups themselves, hyperbolic orientable surface groups, and free products of free abelian groups. A concrete classification is provided by the hierarchy of constructible limit groups.
Definitions and characterizations
The space of marked groups and the Chabauty topology
For n ≥ 1 {\textstyle n\geq 1} , the space of marked groups G n {\textstyle {\mathcal {G}}_{n}} is the set of normal subgroups of the free group F n {\textstyle F_{n}} . Because F n {\textstyle F_{n}} is a discrete group, the Chabauty topology is the topology on G n {\textstyle {\mathcal {G}}_{n}} induced by the product topology, or Tychonoff topology, on the power set { 0 , 1 } F n {\textstyle \{0,1\}^{F_{n}}} (where { 0 , 1 } {\textstyle \{0,1\}} is discrete). Thus one can say that two elements N , N ′ {\textstyle N,N'} of G n {\textstyle {\mathcal {G}}_{n}} are "close" if one has S ∩ N = S ∩ N ′ {\textstyle S\cap N=S\cap N'} for a "big" finite subset S ⊂ F n {\textstyle S\subset F_{n}} . Since a group presentation with n {\textstyle n} generators can be regarded as an epimorphism from F n {\textstyle F_{n}} , which is the same as a quotient of F n {\textstyle F_{n}} , the set of all group presentations involving a set of n {\textstyle n} letters is naturally in bijection with G n {\textstyle {\mathcal {G}}_{n}} and thus inherits its topology. One may regard elements of G n {\textstyle {\mathcal {G}}_{n}} either as subgroups, presentations or epimorphisms. For 1 ≤ k ≤ n {\textstyle 1\leq k\leq n} , a limit group over F k {\textstyle F_{k}} is the quotient of F n {\textstyle F_{n}} by an element of the topological closure of the set of normal subgroups N ◃ F n {\textstyle N\triangleleft F_{n}} such that F n / N {\textstyle F_{n}/N} is isomorphic to F k {\textstyle F_{k}} . As the space G n {\textstyle {\mathcal {G}}_{n}} is compact metrizable, this is the same as a limit of a sequence of epimorphisms ϕ i : F n ⟶ F k {\textstyle \phi _{i}:F_{n}\longrightarrow F_{k}} . A limit group is a finitely generated group for which a presentation arises in this way for some 1 ≤ k ≤ n {\textstyle 1\leq k\leq n} .
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