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Hypercentral unit groups and the hyperbolicity of a modular group algebra

We classify groups G such that the unit group U(ZG) is hypercentral. In the second part we classify groups G whose modular group algebra has hyperbolic unit group V(KG).


Introduction
We denote by Γ = U 1 (ZG) the group of units of augmentation one of the integral group ring ZG of G. Z n (Γ) will denote the n-th centre of Γ e we define Z ∞ (Γ) = n∈N Z n (Γ). An element in Z ∞ (Γ) is called an hypercentral unit. In the finite group case, Arora, Hales and Passi in [1] showed that the central height of Γ is at most 2, that is, Z ∞ (Γ) = Z 2 (Γ). Arora and Passi in [2] then proved that Z ∞ (Γ) = Z(Γ)T , where T denotes the torsion subgroup of Z ∞ (Γ). These results were extended to torsion groups by Li [10] and Li,Parmenter [11]. In [12], [13] they presented some contributions to the problem for non-periodic groups. In [6, Chapter VI] Hertweck extended these results to group rings RG of periodic groups G over G−adapted rings R. From its exposition is clear that the containement of Z ∞ (Γ) in the normalizer N Γ (G) is an important property. We present our contribution to the study of the hypercentral units in [8]. Among many other results in [8] it is proved that the containement of Z ∞ (Γ) in N Γ (G) holds for an arbitrary group G. The support of an hypercentral unit is investigated and it is proved that the normal closure of the group generated by an hypercentral unit is a polycyclic-by-finite group (in case, G is finitely generated).
In [14], Polcino Milies classified finite groups such that the unit group of an integral group ring is nilpotent. This result was extended to arbitrary groups by Sehgal-Zassenhaus in [17]. Since nilpotent groups are hypercentral it is natural to consider the question of classify groups G such that the group of units of an integral group ring U 1 (ZG) is hypercentral. This problem was posed by several leading experts in the field. In the second section we completely solve it as a natural consequence of our research about the hypercentral units of an integral group ring done in [8].
In sections 3, 4, 5 we deal with the topic of hyperbolic unit groups. In the context of hyperbolic unit groups, Juriaans, Passi, Prasad in [9] studied the groups G whose unit group U(ZG) is hyperbolic, classified the torsion subgroups of U(ZG) and the polycyclic-by-finite subgroups of G. We consider the natural question of classifying groups G for which the group of units with augmentation one of a modular group algebra, V (KG), is hyperbolic.
Notation is mostly standard and the reader is referred to [15,16] for general results on group rings. For the theory of hyperbolic groups, we refer the reader to the reference [5].

Groups with Hypercentral Unit Group
Unless otherwise stated explicitly G will always denote an arbitrary group G. Firstly we recall a result proved in [8] which we will need in our investigations. It will also appear in [7].
In particular, elements of Γ that are of finite order and whose order is not a power of 2 commute with Z ∞ (Γ), and Z 2 We need the following result proved in the context of nilpotent unit groups by Sehgal-Zassenhaus in [17].
(1) Every finite subgroup of G is normal in G.
(3) If t has odd order then gt = tg; (4) If 1 = t 1 has odd order, t 2 has even order then T is a central subgroup of G. Proof.
(1) Observe initially that since Γ is hypercentral, we have that G is hypercentral. Since G ⊆ Γ = Z ∞ (Γ) it follows, by Lemma 2.1, that g −1 tg ∈ t , for all g ∈ G, t ∈ T . Since every subgroup of T is normal, T is an abelian subgroup or an Hamiltonian subgroup.
(4) Suppose g ∈ G such that g −1 t 1 g = t 1 and g −1 t 2 g = t −1 2 . It follows that g −1 t 1 t 2 g = t 1 t −1 2 which should be equal to t 1 t 2 or (t 1 t 2 ) −1 . This implies that t 1 t −1 2 = t 1 t 2 and t 2 2 = 1. Hence g −1 t 2 g = t 2 . Finishing the proof. We now state the main result of this section.  Proof.
(⇒) Suppose that Γ = Z ∞ (Γ). By Lemma 2.2 every subgroup of T is normal in G and T is an abelian subgroup or a Hamiltonian subgroup.

Suppose that Γ is hypercentral and T is not central. In any case T is abelian or a Hamiltonian group.
Suppose firstly that T is a non-central Hamiltonian group with an element x of odd order. Consider the subgroup H = K 8 × x . In this case it follows by item (4) of Lemma 2.2 that H is a central subgroup of G. Contradiction.
Suppose that T is an abelian, non-central subgroup of G and that g −1 t 1 g = t 1 , g −1 t 2 g = t −1 2 for some t 1 , t 2 ∈ T . Then g −1 t 1 t 2 g = t 1 t −1 2 . But by Lemma 2.1, g −1 t 1 t 2 g must be equal to either t 1 t 2 or t −1 1 t −1 2 . Hence either t 2 1 = 1 and g −1 t i g = t −1 i , i = 1, 2 or t 2 2 = 1 and g −1 t i g = t i . In any case we obtain that g −1 t 1 g = t δ(g) 1 . Also T (G) is an abelian 2−group by Lemma 2.2. Denote by K 8 = i, j | i 2 = j 2 = u, u 2 = 1, ji = iju . Every g ∈ G maps every subgroup of K 8 onto itself and induces the identity in K 8 / i 2 . In fact only one of the four inner automorphisms (⇐) Under the hypothesis of the Theorem we must prove that Γ is hypercentral. Since G/T is an ordered group and QT have no nilpotent elements, it follows by [16,Theorem 45.7] that Γ = U 1 (ZT )G.
We must consider three cases separately.
To see this let u = g∈G α g g ∈ U 1 (ZT ), x ∈ G. So u x = g∈G α g x −1 gx. Since x centralizes the elements of T or x acts by inversion on the elements of T we obtain in the first case that u x = u, γ = 1 and in the second case we obtain that u x = u * , γ = u −1 u * . Since T is abelian, it follows that γ * = γ −1 and by [16,Proposition 1.3] γ = ±t, for some t ∈ T . Since γ has augmentation 1, γ ∈ T . We conclude that Continuing this process and using the fact that G is a hypercentral group, we conclude that U 1 (ZT ) ⊆ Z ∞ (Γ) and Γ is hypercentral.
(3) Suppose (c). In this case, since T is an Hamiltonian 2−group. It is well known that in this case U 1 (ZT ) has only trivial units. It follows that U 1 (ZT ) = T . Consequently, Γ = G is hypercentral.

Modular Group Algebras with Hyperbolic V (KG)
Let Z 2 denote the free Abelian group of rank two, p be a rational prime, GF (p n ) will denote the Galois Field with p n elements, tr.deg(K) denotes the transcendence degree of the field K over GF  In what follow we investigate under which conditions the group of units of a modular group algebra of an arbitrary (non-trivial) group G is hyperbolic.
We denote by J (KG) the Jacobson Radical of KG and ω(G) represents the augmentation ideal of KG.
Lemma 3.2. Suppose that G is a finite (non-trivial) group and K is a field, char(K) = p > 0, tr.deg(K) ≥ 1. Then V (KG) is not hyperbolic. Theorem 3.4. Let G be an arbitrary group with torsion, K be a field, char(K) = p > 0. If V (KG) is hyperbolic then K is algebraic over GF (p).
Our next Theorem considers the case in which G is an arbitrary (non-trivial) group, K a field of char(K) = p > 0 under the hypothesis that U(KG) is hyperbolic.
Theorem 3.5. Let G be an arbitrary (non-trivial) group, K a field of char(K) = p > 0. If U(KG) is hyperbolic then K is finite.