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Makanin-Razborov diagrams for limit groups Emina Alibegovi c
 

Summary: Makanin-Razborov diagrams for limit groups
Emina Alibegovic
September 27, 2004
Abstract
We give a description of Hom(G;L), where L is a limit group
(fully residually free group). We construct a nite diagram of groups,
Makanin-Razborov diagram, that gives a convinient representation of
all such homomorphisms.
1 Introduction
The subject of this paper has the roots in the following problem: given a
nitely presented group G, describe the set of all homomorphisms G ! F to
a xed free group F .
When G is the fundamental group of a closed surface, say of genus g,
Stallings answered our question as follows. Denote by q : G ! F g an epi-
morphism to the free group of rank g (de ned by inclusion of the boundary
to the handlebody). Then every f : G ! F factors as f =  Ć q Ć , for
some automorphism : G ! G, and some  : F g ! F . Thus Hom(G;F )
is `parametrized' by the product of the Teichmuller modular group of G and
the `aĆne space' F g . This theorem of Stallings was generalized to arbitrary
nitely generated groups G by Sela in [14] and Kharlampovich and Myas-

  

Source: Alibegovic, Emina - Department of Mathematics, University of Utah

 

Collections: Mathematics