Makanin–Razborov diagrams for hyperbolic groups

Abstract

We give a detailed account of Zlil Sela's construction of Makanin-Razborov diagrams describing Hom(G, Γ) where G is a finitely generated group and Γ is a hyperbolic group. We also deal with the case where Γ has torsion. Les diagrammes Makanin-Razborov pour les groupes hyperboliques Résumé Nous proposons une présentation détaillée de la construction des diagrammes de Makanin-Razborov de Zlil Sela qui décrivent Hom(G, Γ) pour un groupe G de type fini et un groupe hyperbolique Γ. De plus, nous traitons le cas où Γ est un groupe ayant de la torsion.

Homomorphisms to 3-manifold groups

“…There are some additional complications in [12,Theorem 5.23] which do not arise here, but the differences between the proofs are almost all notational. Similar arguments also appear in the proof of [36,Proposition 4.17 ].…”

“…Any element of g can be represented (non-uniquely) by [a 0 , e 1 , a 1 , ..., e n , a n ] where e 1 , ..., e n is an edge path in A from v 0 to v 0 , a 0 ∈ A v 0 , and each a i ∈ A v i where v i is the terminal vertex of e i for 1 ≤ i ≤ n. Suppose v is a vertex of A and σ ∈ Aut(A v ) acts by conjugation on each adjacent edge group. Then σ can be extended to an automorphism of G as in [36,Definition 4.13] as follows: for each adjacent edge e there is γ e ∈ A v so that σ (h) = γ e hγ −1 e for all h in the image of A e in A v . Then σ extends to σ ∈ Aut(H) by defining σ ([a 0 , e 1 , a 1 , ..., e n , a n ]) = [a 0 , e 1 , a 1 , ..., e n , a n ] where…”

“…As in [36] (see also [12]), we create a sequence of graphs of groups approximating J so that the φ i factor through the terms in the approximating graphs of groups. This is essentially the same as [36,Lemma 7.1], and the proof from there works in our situation without change. The idea comes from [29].…”

Formal solutions and the first‐order theory of acylindrically hyperbolic groups

“…Recall that a group is said to be equationally Noetherian if the set of solutions of any system of equations in finitely many variables coincides with the set of solutions of a certain finite subsystem of this system. As a consequence of the Hilbert Basis Theorem, linear hyperbolic groups are equationally Noetherian, and it was proved by Sela in [37] (torsion‐free case) and by Reinfeldt and Weidmann in [29] (general case) that the linearity assumption can be dropped. Equational Noetherianity has proved extremely useful in the study of the first‐order theory of hyperbolic groups, notably because limit groups over hyperbolic groups are not finitely presentable in general, which constrains us to deal with infinite systems of relations.…”

“…We overcome this problem by introducing a method of approximating, in a precise sense, limit groups over acylindrically hyperbolic groups by finitely presented groups relative to a subgroup. The idea of approximating limit groups by finitely presented groups already appears, in a slightly different form, in [34, Theorem 3.2] (see also [17; 18, Lemma 6.3; 29, Lemma 6.1]). More precisely, in the present case, there exists a quotient $$A$$ of $$\Gamma _{\Sigma ,\bm {\gamma }}$$, called an approximation of $$L$$, which is finitely presented relative to $$G$$, maps onto $$L$$, and has a splitting that mimics the splitting of $$L$$ outputted by the Rips machine.…”