Rational group ring elements with kernels having irrational dimension

Austin, Tim

doi:10.1112/plms/pdt029

Cited by 28 publications

(47 citation statements)

References 17 publications

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“…for any x, y ∈ X. Thus for any ε > max j>k λ j diam( [3,9,12,13,15,25,40] Coornaert and Krieger showed that if a countable amenable group Γ has subgroups of arbitrary large finite index, then for any r ∈ [0, +∞] there is a continuous action of Γ on some compact metrizable space X with mdim(X) = r [8]. Thus it is somehow surprising that the value of the mean dimension of algebraic actions of some amenable groups is rather restricted, as the following consequence of Theorems 1.1 and 8.3 shows: (see also [7] for some related discussion) We shall answer Question 9.1 for algebraic actions of the groups in Theorem 8.3.…”

Section: For a Continuous Action γmentioning

confidence: 99%

Mean dimension, mean rank, and von Neumann–Lück rank

Liang

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We introduce an invariant, called mean rank, for any module{\mathcal{M}}of the integral group ring of a discrete amenable group Γ, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced Γ-action on the Pontryagin dual of{\mathcal{M}}, the mean rank of{\mathcal{M}}, and the von Neumann–Lück rank of{\mathcal{M}}all coincide. As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin–Schnirelmann theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of finite subgroups, algebraic actions with zero mean dimension are inverse limits of finite entropy actions.

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Section: For a Continuous Action γmentioning

confidence: 99%

Mean dimension, mean rank, and von Neumann–Lück rank

Liang

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…This was first observed by L. Grabowski and B. Virag in [12,15,26] who used this connection to show that the spectral measure associated with the operator H β of convolution with the element a+a −1 +βc ∈ R[L] is singular continuous. This was achieved through a modification of a method from [1], which appeared in an unpublished work of Grabowski and Virag [15] and for which an exposition appears in the appendix of [13]. The idea is to use an application of the Fourier transform using Pontryagin's dual A = Z/2Z of the base A = Z Z/2Z of the wreath product where {u(n)} n∈Z ∈ 2 (Z) and ω is a sequence of i.i.d random variables taking values in {0, 1} with Bernoulli distribution the assigns probability 1/2 to 0 and 1.…”

Section: The Relation With the Theory Of Random Jacobi And Schrödinger mentioning

confidence: 99%

“…Let {H ω } ω∈Ω be a random ergodic family of Jacobi operators on Z of the form (1) above. An interesting question that is related to this discussion is about the decomposition of the representation π n into irreducible components.…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

Spectra of Cayley graphs of the lamplighter group and random Schrödinger operators

Grigorchuk¹,

Simanek²

2021

Trans. Amer. Math. Soc.

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We show that the lamplighter group L = Z/2Z Z has a system of generators for which the spectrum of the discrete Laplacian on the Cayley graph is a union of an interval and a countable set of isolated points accumulating to a point outside this interval. This is the first example of a group with infinitely many gaps in the spectrum of Cayley graph. The result is obtained by a careful study of spectral properties of a one-parametric family a + a −1 + b + b −1 − µc of convolution operators on L where µ is a real parameter.Our results show that the spectrum is a pure point spectrum for each value of µ, the eigenvalues are solutions of algebraic equations involving Chebyshev polynomials of the second kind, and the topological structure of the spectrum makes a bifurcation when parameter µ passes the points 1 and −1. Namely if |µ| ≤ 1 the spectrum is the interval while when |µ| > 1 it is a union of the interval and a countable set of points accumulating to a point outside the interval.

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“…Of course, non-integral examples are well-known and can be easily constructed for any group with torsion. Irrational examples were first constructed by T. Austin [4]. Additional examples appear in [8,18].…”

Section: Introductionmentioning

confidence: 99%