Primitivity of group rings of non-elementary torsion-free hyperbolic groups

TL;DR

This paper proves that group rings of non-elementary torsion-free hyperbolic groups over fields or countable domains are primitive, extending understanding of their algebraic structure.

Contribution

It establishes the primitivity of group rings for a broad class of hyperbolic groups using recent theoretical results.

Findings

Group rings over fields are primitive.

Primitivity holds for countable domain coefficients.

Extends algebraic understanding of hyperbolic groups.

Abstract

We use a recent result of Alexander and Nishinaka to show that if $G$ is a non-elementary torsion-free hyperbolic group and $R$ is a countable domain, then the group ring $RG$ is primitive. This implies that the group ring $KG$ of any non-elementary torsion-free hyperbolic group $G$ over a field $K$ is primitive.

Equations

$$u^{n_0}{g}_1{u}^{n_1}{g}_2\cdots u^{n_{k-1}}{g}_k{u}^{n_k}\ne 1.$$

$$u^{n_0}{g}_1{u}^{n_1}{g}_2\cdots g_{m-1}{u}^{n_{m-1}}{g}_m{u}^{n_m}\ne 1$$

$$w=(x_1^{-1}{g}_1{x}_1)(x_2^{-1}{g}_2{x}_2)\cdots (x_m^{-1}{g}_m{x}_m)$$

$$w=u^{n_0}{g}_1{u}^{n_1}{g}_2\cdots g_{m-1}{u}^{n_{m-1}}{g}_m{u}^{n_m},$$

Full text

1 Introduction

In [1], Alexander and Nishinaka use the following Property (*) of a group $G$ to establish the primitivity of a broad class of group rings. (Recall that a ring $R$ is (right) primitive if it contains a faithful irreducible (right) $R$-module.)

(*) For each subset $M$ of $G$ consisting of a finite number of elements not equal to 1, and for any positive integer $m\geq 2$, there exist distinct $a,b,c\in G$ so that if $(x_{1}^{-1}g_{1}x_{1})(x_{2}^{-1}g_{2}x_{2})\cdots(x_{m}^{-1}g_{m}x_{m})=1$, where $g_{i}\in M$ and $x_{i}\in\{a,b,c\}$ for all $i=1,\dots,m$, then $x_{i}=x_{i+1}$ for some $i$.

In particular, Alexander and Nishinaka give the following broad criterion for primitivity:

Theorem 1** ([1, Theorem 1.1]).**

Let $G$ be a group which has a non-Abelian free subgroup whose cardinality is the same as that of $G$, and suppose that $G$ satisfies Property (). Then, if $R$ is a domain with $|R|\leq|G|$, the group ring $RG$ of $G$ over $R$ is primitive. In particular, the group algebra $KG$ is primitive for any field $K$.*

Our goal at present is to show that the hypotheses of Theorem 1 hold if $G$ is a non-elementary torsion-free hyperbolic group, thus implying the primitivity of a fairly broad class of group rings.

2 Hyperbolic groups

Let $G$ be a group with finite generating set $X$. Recall that the Cayley graph $\Gamma_{X}(G)$ of $G$ with respect to $X$ is an $X$-digraph with vertex set $G$ and an $x$-labelled edge directed from $g$ to $gx$ for all $g\in G$ and $x\in X$. We may equip $\Gamma_{X}(G)$ with the word metric by assigning each edge a length of one; as a result, $\Gamma_{X}(G)$ becomes a geodesic metric space.

We say that $\Gamma_{X}(G)$ satisfies the $\delta$-thin triangles condition for a real constant $\delta>0$ if every side of a geodesic triangle is contained in the $\delta$-neighborhood of the union of the two remaining sides. If a group $G$ admits a finite generating set $X$ and real constant $\delta>0$ for which $\Gamma_{X}(G)$ satisfies the $\delta$-thin triangles condition, we say that $G$ is $\delta$-hyperbolic, or simply hyperbolic. Hyperbolicity does not depend on choice of finite generating set, although the constant of hyperbolicity may vary.

It is easy to see that finite groups and free groups are hyperbolic. Some more interesting examples of hyperbolic groups include one-relator groups with torsion, certain small cancellation groups, and fundamental groups of closed surfaces with negative Euler characteristic. On the other hand, free Abelian groups of rank at least two and Baumslag-Solitar groups are standard examples of non-hyperbolic groups.

The intrinsic geometry of hyperbolic groups induces a wide range of algebraic, geometric, and algorithmic properties. Hyperbolic groups are necessarily finitely presented, have a solvable word problem, have linear isoperimetric inequality, and satisfy the Tits alternative. Furthermore, hyperbolicity is a common phenomenon among groups: standard statistical models in group theory show that a randomly chosen finitely presented group is overwhelmingly likely to be infinite, torsion-free, and hyperbolic [9]. Hyperbolic groups and their generalizations are the subject of a tremendous amount of modern literature in group theory, and so we refer the interested reader to [2, 6, 7] for a more thorough introduction.

We say that a hyperbolic group is elementary if it is finite or virtually infinite cyclic; in either case, primitivity of a group ring over such a group has already been characterized. Fundamental to these characterizations is the FC center of a group $G$, the set of elements $\Delta(G)=\{g\in G\mid[G:C_{G}(g)]<\infty\}$. Similarly, the FC+ center of $G$ is the subset $\Delta^{+}(G)$ of torsion elements of $\Delta(G)$.

It is well-known that if $KG$ is primitive, then $\Delta^{+}(G)=1$ [11, Lemma 9.1.1]. Consequently, if $G$ is finite, then $KG$ is not primitive for any field $K$ unless $G=1$. Alternately, if $G$ is virtually infinite cyclic, it is necessarily polycyclic-by-finite and therefore covered by the classification collectively due to Domanov [4], Farkas-Passman [5], and Roseblade [12]: for any field $K$ and polycyclic-by-finite $G$, the group ring $KG$ is primitive if and only if $\Delta(G)=1$ and $K$ is not absolute.

We now restrict our attention to non-elementary torsion-free hyperbolic groups. We begin with a theorem of Gromov.

Theorem 2** ([7, Theorem 5.3.E]).**

There exists a constant $E=E(k,\delta)>0$ such that for every $k$ elements $g_{1},\dots,g_{k}$ of infinite order in a $\delta$-hyperbolic group $G$, the subgroup $\langle g_{1}^{e_{1}},\dots,g_{k}^{e_{k}}\rangle$ is free whenever $e_{i}\geq E$ for $i=1,\dots,k$.

We obtain the following immediate corollary.

Corollary 3**.**

Let $G$ be a non-elementary torsion-free hyperbolic group. Let $M=\{g_{1},\dots,g_{k}\}$ be a finite subset of nonidentity elements of $G$. Then there exists an element $u\in G$ which is not a proper power and which commutes with no element of $M$.

Proof.

Let $C(g)=C_{G}(g)$ denote the centralizer of $g$ in $G$. It is well-known that centralizers of nontrivial elements of a torsion-free hyperbolic group are necessarily infinite cyclic [6]. Consequently, if $g\in G$ is nontrivial and not a proper power, then $C(g)=C(g^{n})=\langle g\rangle$ for all $n\neq 0$ and $C(g)$ is maximal among the cyclic subgroups of $G$. Thus we may replace $g_{i}^{n}\in M$ with $g_{i}$ without changing the set of elements which commute with no element of $M$, and we may assume that no element of $M$ is a proper power. The same argument shows that we may always produce $u$ which is not a proper power by passing to a root if necessary.

Suppose $k=1$. As $G$ is non-elementary, it must properly contain the maximal cyclic subgroup $C(g_{1})$, and so there must exist an element $u$ of $G$ failing to commute with $g_{1}$.

Now suppose $k\geq 2$. Let $E=E(k,\delta)$ be the constant from Theorem 2; we may assume $E$ to be a positive integer. Since $G$ is torsion-free, $g_{i}^{E}$ is nontrivial and of infinite order for all $g_{i}\in M$. Therefore $H=\langle g_{1}^{E},\dots,g_{k}^{E}\rangle$ is a non-Abelian free subgroup of $G$ and so contains an element $u\in H\subseteq G$ which fails to commute with any $g_{i}^{E}$. Since $C(g_{i}^{E})=C(g_{i})$ in $G$, we have that $u$ cannot commute with any $g_{i}$. ∎

In order to show that a non-elementary torsion-free hyperbolic group satisfies Property (*), we resort to a version of the big powers property of torsion-free hyperbolic groups. The form we use here follows immediately from the stronger version for a class of relatively hyperbolic groups proved in [8].

Theorem 4** (The big powers property [8]).**

Let $G$ be a torsion-free hyperbolic group. Let $u\in G$ be nontrivial and not a proper power. Let $g_{1},\dots,g_{k}$ be elements of $G$ which do not commute with $u$. Then there exists $N>0$ such that if $|n_{i}|\geq N$ for $i=0,\dots,k$ then

[TABLE]

The big powers property provides a way of generating a large set of nontrivial elements of a group and so is of use in studying residual properties of groups, universal equivalence, and algebraic geometry over groups. The property appears first due to B. Baumslag in his study of fully residually free groups [3]. Ol’shanskiĭ later generalizes the property to torsion-free hyperbolic groups [10], and Kharlampovich and Myasnikov further generalize it to non-Abelian torsion-free relatively hyperbolic groups with free Abelian parabolic subgroups [8].

3 Main Result

Proposition 5**.**

If $G$ is a non-elementary torsion-free hyperbolic group, then $G$ satisfies Property ().*

Proof.

Let $M$ be a finite subset of $G$ not containing the identity. By Corollary 3, there exists an element $u\in G$ which generates its own centralizer and commutes with no $g\in M$.

Fix a positive integer $m\geq 2$ and consider a finite sequence $g_{1},\dots,g_{m}$ of elements from $M$. Since $u$ commutes with none of the $g_{i}$ and $u$ generates its own centralizer, by the big powers property, there exists $N(g_{1},\dots,g_{m})>0$ such that

[TABLE]

whenever $|n_{i}|\geq N$ for all $i=0,\dots,m$.

Since $M$ is a finite set, there are finitely many $m$-tuples $(g_{1},\dots,g_{m})$ drawn from $M$. Therefore, let $N>\max\left\{N(g_{1},\dots,g_{m})\mid g_{1},\dots,g_{m}\in M\right\}$.

We now define $a=u^{N},b=u^{2N}$, and $c=u^{3N}$. Since $G$ is torsion-free, these elements are necessarily distinct. Consider a product

[TABLE]

where $x_{1},\dots,x_{m}\in\{u^{N},u^{2N},u^{3N}\}$. We then have

[TABLE]

where $u^{n_{0}}=x_{1}^{-1},u^{n_{m}}=x_{m},u^{n_{i}}=x_{i}x_{i+1}^{-1}$ and $n_{i}\in\{0,\pm N,\pm 2N\}$ for $i=1,\dots,m-1$. Note that by choice of $x_{1}$ and $x_{m}$, we have $n_{0}\neq 0$ and $n_{m}\neq 0$.

By the big powers property and choice of $N$, if $n_{i}\neq 0$ for all $i=0,\dots,m$, then $w\neq 1$. Therefore, if $w=1$, then some $n_{i}=0$. Since we cannot have $n_{0}=0$ or $n_{m+1}=0$, we have $n_{i}=0$ for some $i\in\{1,\dots,m-1\}$, in which case we must have $1=u^{n_{i}}=x_{i}x_{i+1}^{-1}$, and so $x_{i}=x_{i+1}$. ∎

Since a non-elementary hyperbolic group is finitely generated by definition, it is necessarily countably infinite. However, by [1, Remark 3.6], a countably infinite group satisfying Property () also necessarily contains a free subgroup of rank two, which is also countably infinite. (We also note that a group $G$ satisfying Property () also automatically satisfies $\Delta(G)=1$.) A non-elementary torsion-free hyperbolic group thus satisfies Property (*) and has a free subgroup of the same cardinality, and so we obtain the following main result as a corollary to Theorem 1.

Theorem 6**.**

If $G$ is a non-elementary torsion-free hyperbolic group, then for any countable domain $R$, the group ring $RG$ of $G$ over $R$ is primitive. In particular, the group ring $KG$ is primitive for any field $K$.

We may slightly relax the torsion-free condition on $G$ if $\Delta(G)=1$, as demonstrated by the following corollary.

Corollary 7**.**

If $G$ is a non-elementary virtually torsion-free hyperbolic group with $\Delta(G)=1$, then for any countable domain $R$, the group ring $RG$ of $G$ over $R$ is primitive. In particular, the group ring $KG$ is primitive for any field $K$.

Proof.

Let $K$ be any field and let $H$ be a torsion-free subgroup of finite index in $G$. As a finite index subgroup, $H$ is quasi-isometric to $G$, and thus $H$ is necessarily also non-elementary and hyperbolic [7]. By Theorem 6, we have that $KH$ is primitive.

From [11, Theorem 4.2.10], KG is prime if and only if $\Delta(G)$ is torsion-free Abelian. Thus if $\Delta(G)=1$, we have that $KG$ is necessarily prime.

Finally, if $KG$ is prime, then $KG$ is primitive if and only if $KH$ is primitive by a result of Rosenberg [13, Theorem 3] (cf. [11, Theorem 9.1.11].) ∎

It is worth noting the long-standing open conjecture that every hyperbolic group is virtually torsion-free. An affirmation of this conjecture, together with the above corollary, would imply that $KG$ is primitive for any field $K$ and any non-elementary hyperbolic group $G$.

Acknowlegements

The author would like to thank Prof. Tsunekazu Nishinaka for introducing the author to the subject of primitive group rings and for many illuminating conversations, Prof. Hisaya Tsutsui for his support and guidance, and the referee for his or her very helpful feedback.