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

We show that the lamplighter group L 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 its Cayley graph. The result is obtained by a careful study of spectral properties of a one-parametric family of convolution operators on L. Our results show that the spectrum is a pure point spectrum for each value of the parameter, 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 the parameter passes the points 1 and -1

Lie Algebras and Growth in Branch Groups

We compute the structure of the Lie algebras associated to two examples of branch groups, and show that one has finite width while the other, the ``Gupta-Sidki group'', has unbounded width. This answers a question by Sidki. More precisely, the Lie algebra of the Gupta-Sidki group has Gelfand-Kirillov dimension $\log3/\log(1+\sqrt2)$. We then draw a general result relating the growth of a branch group, of its Lie algebra, of its graded group ring, and of a natural homogeneous space we call "parabolic space", namely the quotient of the group by the stabilizer of an infinite ray. The growth of the group is bounded from below by the growth of its graded group ring, which connects to the growth of the Lie algebra by a product-sum formula, and the growth of the parabolic space is bounded from below by the growth of the Lie algebra. Finally we use this information to explicitly describe the normal subgroups of the "Grigorchuk group". All normal subgroups are characteristic, and the number of normal subgroups of index $2^n$ is odd and is asymptotically $n^{\log_2(3)}$

On the reversibility and the closed image property of linear cellular automata

When $G$ is an arbitrary group and $V$ is a finite-dimensional vector space, it is known that every bijective linear cellular automaton $\tau \colon V^G \to V^G$ is reversible and that the image of every linear cellular automaton $\tau \colon V^G \to V^G$ is closed in $V^G$ for the prodiscrete topology. In this paper, we present a new proof of these two results which is based on the Mittag-Leffler lemma for projective sequences of sets. We also show that if $G$ is a non-periodic group and $V$ is an infinite-dimensional vector space, then there exist a linear cellular automaton $\tau_1 \colon V^G \to V^G$ which is bijective but not reversible and a linear cellular automaton $\tau_2 \colon V^G \to V^G$ whose image is not closed in $V^G$ for the prodiscrete topology