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Is the property of having a bound on the dimensions of irreducible representations preserved by an extension?

For example $G_1=\mathbb{Z}$ and $G_2=\mathbb{Z}/2\mathbb{Z}$ are discrete abelian groups each of which only have 1 dimensional irreducible representations, while the group $$\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z}$$ has only irreducible representations of dimensions one and two.

One of the purposes of this question for me is to get a feel for how difficult problems of this kind are as I really do not know very much about unitary representations of groups.

The case when the groups considered are finitely generated discrete (elementary) amenable would be very helpful to me.

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Are you only considering finite-dimensional unitary representations? – Qiaochu Yuan Aug 10 '12 at 4:26
(If you are considering infinite-dimensional representations then I am not sure what you mean by "bound." Certainly the dimension of an irreducible representation of $G$ is bounded by $|G|$...) – Qiaochu Yuan Aug 10 '12 at 4:32
@Qiaochu: I guess that part of the hypothesis is that all unitary irreps of $G$ are finite-dimensional. See my answer below. – Yemon Choi Aug 10 '12 at 4:37
Very helpful and very fast, thank you! – Michael Sun Aug 10 '12 at 7:06
@MIchael: just lucky (for some, at least) that I was working late in the office and have had similar questions on my mind. – Yemon Choi Aug 10 '12 at 8:14
up vote 13 down vote accepted

A discrete group has the property you describe if and only if it is virtually abelian. I must admit to never learning the proof, but some foraging on MathSciNet indicates the result is due to Isaacs and Passman:

Isaacs, I. M.; Passman, D. S. Groups with representations of bounded degree. Canad. J. Math. 16 1964 299--309 | MathReview

(I seem to remember hearing that there are proofs which get better estimates on the degree of the abelian subgroup, but I may be misremembering.)

The class of virtually abelian groups has good hereditary properties. However, it is not stable under semi-direct products: indeed there are metabelian groups, arising as crossed products, which are not virtually abelian, e.g. integer Heisenberg, or the ax+b group over the rationals. Indeed, in the former case once can write down infinite-dimensional unitary irreps. So I think you need to restrict the kinds of extension you are considering if you want this property to be preserved.

If you are interested in the problem beyond the discrete setting, then I think one place to start would be with the paper of C. C. Moore

Calvin C. Moore, Groups with finite dimensional irreducible representations. Trans. Amer. Math. Soc. 166 (1972), 401--410. | MathReview

(A locally compact group $G$ is called a Moore group if all its (continuous) unitary irreps are finite-dimensional. It is a hard result of Thoma that if a discrete group is Moore then there is a uniform bound on the degrees of the irreps, which is the property described in your question. Of course non-discrete compact groups show that the two concepts are in general distinct.)

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This property is not preserved under extension. The discrete Heisenberg group $H_3(\mathbb{Z})$, which consists of integer matrices of the form

$$\left[ \begin{array}{ccc} 1 & a & b \\\ 0 & 1 & c \\\ 0 & 0 & 1 \end{array} \right],$$

is a central extension of $\mathbb{Z} \times \mathbb{Z}$ by $\mathbb{Z}$. It has finite quotients $H_3(\mathbb{F}_p)$ with irreducible representations of dimension $p$ (this follows from the fact that $H_3(\mathbb{F}_p)$ is non-abelian and has order $p^3$).

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