On the linearity of HNN-extensions with abelian base

Abstract

We show that an HNN-extension with finitely generated abelian base group is Z-linear if and only if it is residually finite.

Linearity of graph products

“…Suppose the result is satisfied for n ą 0. By Lemma 4.4 we know that Ip kq is a precycle, which together with the induction hypothesis Ip kq X p Ť n i"0 L i q " K n and Lemma 3.6 implies that Ip kq X L n`1 Ď K n`1 , and therefore that (10) Ip kq X L n`1 Ď M n`1 " K n`1 X L n`1 .…”

“…Over the course of the years, results concerning free products and amalgamated products [9,11,12,14], and HNN extensions [10], of linear groups have appeared. Shalen [12] in particular proved that amalgamated products of linear groups over C, amalgamated over maximal cyclic subgroups, are linear over Cptq, where t is an indeterminate, and therefore also linear over C (compare [12,Lemma 1.2], or Lemma 2.8), and that the class of linear groups over C is preserved under taking free products (compare [12,Theorem 1]).…”

“…On the other hand, HNN extensions of free abelian groups [10] are linear over Z. Moreover, it is known that graph products of some specific families of linear groups are again linear over Z: any graph product of subgroups of Coxeter groups [6, Theorem 3.2], and in particular any right-angled Artin group [7] is linear over Z.…”

Tubular groups, 1-relator groups and nonpositive curvature

“…In this regard, we note the conjecture of Metaftsis, Raptis and Varsos in [20] that the fundamental group of a graph of groups, where the graph is a finite tree and the vertex groups are all finitely generated abelian, is linear. Thus this result establishes their conjecture in the case of Z 2 vertex groups and Z edge groups.…”

The Formanek-Procesi group with base a right-angled Artin group: Residual nilpotence and Lie algebra

“…It was shown in [9] that F P (F n ) is not linear. On the other hand, it was shown in [15] that an HNN-extension with base a finitely generated abelian group is linear if and only if is residually finite. Hence, if A is a finitely generated abelian group, then F P (A) is an HNN-extension with base an abelian group and, by results of [1], it is residually finite and hence, it is linear.…”