Limit groups as limits of free groups: compactifying the set of free groups.

By Christophe Champetier, Vincent Guirardel. 2004

69 citations recorded by the source corpus.

We give a topological framework for the study of Sela's limit groups:limit groups are limits of free groups in a compact space of markedgroups.Many results get a natural interpretation in this setting.The class of limit groups is known to coincide with the class offinitely generated fully residually free groups.The topological approach gives some new insight on the relation betweenfully residually free groups, the universal theory of free groups,ultraproducts and non-standard free groups.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/be792842ddedfbf1c524de785e6b0ec73ffb9df5

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Krull dimension of solvable groups (2008, background): “The unification theorem mentioned above implies that the coordinate groups of irreducible varieties in rigid groups are also rigid, as well as the limits of rigid groups in the Gromov-Hausdorff topology [21], see also [20], and more details in [10, 11].”
Orderable 3-manifold groups (2002, background): “(Another approach to the bi-orderability of surface groups can be found in a recent paper of Champetier and Guirardel [ CR ])”
On the isolated points in the space of groups (2005, background): “Various elementary observations in these directions are made in [CG05]: for instance, the class of nilpotent finitely generated groups is open, while the class of solvable finitely generated groups is not; the class of finitely generated orderable groups is closed, etc.”
Hall’s Theorem for Limit Groups (2006, methodology): “Another proof using different techniques is given in [5].”
Unification theorems in algebraic geometry (2008, background): “…varieties over G [4, 24, 25, 26, 47], groups universally equivalent to G [40, 13, 34], limit groups of G in the Grigorchuk-Gromov’s metric [10], in the theory of equations in groups [28, 38, 39, 24, 25, 26, 18], in group actions [8, 12, 35, 17, 14], in the solutions of Tarski problems…”
Subgroups of direct products of limit groups (2007, result): “ [3]. The papers of Kharlampovich and Myasnikov [13, 14] approach the subject from a perspective more in keeping with traditional combinatorial group theory, and a further perspective is developed in [11]. 2.1 Limit groups A ﬁnitely generated group Γ is a limit group if for every ﬁnite set T ⊂ Γ the”