nLab fully residually free group

Definition

A (discrete) group $G$ is calld fully residually free if for each finite subset $S\subset G$ there exists a free group $F_S$ and a homomorphism of groups $f \colon G\to F_S$ such that $f|_S$ is injective (images of elements of $S$ under $f$ are pairwise distinct).

A fully residually free group which is finitely generated is also called a limit group.

Properties

(describe limit groups via limit of free group presentations in a specific topology)

Applications

Limit groups play a major role in the study of equations over free groups and corresponding Makanin–Razborov diagrams.

Related entries

• Makanin-Razborov diagram

References

• Zlil Sela: Diophantine geometry over groups. I. Makanin–Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. 93 (2001) 31–105 [doi:10.1007/s10240-001-8188-y, numdam:PMIHES_2001__93__31_0]

• Wikipedia: Limit group

• Emina Alibegović, Mladen Bestvina: Limit groups are $CAT(0)$, J. London Math. Soc. 74 2 (2006) 259–272 [doi:10.1112/S0024610706023155]

• M. Bestvina, M. Feighn: Notes on Sela’s work: limit groups and Makanin–Razborov diagrams, in: Geometric and Cohomological Methods in Group Theory, LMS Lecture Note Series, 358 (2009) 1-29 [arXiv:0809.0467]