Virtual Retraction Properties in Groups

Abstract

If $G$ is a group, a virtual retract of $G$ is a subgroup, which is a retract of a finite index subgroup. Most of the paper focuses on two group properties: property (LR), that all finitely generated subgroups are virtual retracts; and property (VRC), that all cyclic subgroups are virtual retracts. We study the permanence of these properties under commensurability, amalgams over retracts, graph products, and wreath products. In particular, we show that (VRC) is stable under passing to finite index overgroups, while (LR) is not. The question whether all finitely generated virtually free groups satisfy (LR) motivates the remaining part of the paper, studying virtual free factors of such groups. We give a simple criterion characterizing when a finitely generated subgroup of a virtually free group is a free factor of a finite index subgroup. We apply this criterion to settle a conjecture of Brunner and Burns.

The profinite completion of relatively hyperbolic virtually special groups

“…Then by [8,Theorem 1.4] G contains F × Z, where F is free non-abelian. Let H be the closure F × Z in G. Since the center of F × Z is cyclic and cyclic groups of virtually compact special groups are virtual retracts (see [19,Corollary 1.6]) and so are separable, the center Z(H) of H contains Z. By Lemma 3.6 H contains a free non-abelian pro-p subgroup.…”

“…We first observe that the profinite version of Hyperbolization theorem [33, Theorem A] extends to virtually compact special groups using [8, Corollary 1.2], [19,Corollary 1.6] and [33,Theorem F].…”

Commensurating HNN extensions: nonpositive curvature and biautomaticity

“…). Now, by [26,Lemma 4.2], there exists a normal subgroup R H which intersects N trivially and such that |H : N R| < ∞. Let H 1 := H/R and φ : H × F → H 1 × F be the natural homomorphism whose kernel is R.…”

Aritmethic lattices of $\SO(1,n)$ and units of group rings

“…This property was first introduced by Long and Reid in [22], however, implicitly they were investigated much earlier. Many important groups possesses this property: free groups, surface groups, hyperbolic 3-manifold groups, virtually special groups, Right angled Artin groups and Coxeter groups (see [25,Corolary 1.6]). In fact, Minasyan proved [25, Theorem 1.5] that VCR is equivalent to virtual abelian retract property.…”

“…Note that VCR is stable under direct product and commensurability (see [25,Theorem 1.4] that allows us to deduce the following characterization: Theorem 1.4. Let G be a finite group and ZG be its group ring.…”