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Register a new userA property of the lamplighter group
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We show that the inert subgroups of the lamplighter group fall into exactly five commensurability classes. The result is then connected with the theory of totally disconnected locally compact groups and with algebraic entropy.
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We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute $ell^2$-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational $ell^2$-Betti numbers arising from the lamplighter group algebra $K[mathbb{Z}_2 wr mathbb{Z}]$, being $K$ a subfield of the complex numbers closed under complex conjugation. This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational $ell^2$-Betti numbers from the algebras $mathbb{Q}[mathbb{Z}_n wr mathbb{Z}]$, where $n geq 2$ is a natural number. We also apply the techniques developed to the (generalized) odometer algebra $mathcal{O}(overline{n})$, where $overline{n}$ is a supernatural number. We compute its $*$-regular closure, and this allows us to fully characterize the set of $ell^2$-Betti numbers arising from $mathcal{O}(overline{n})$.
The goal of this paper is to give a group-theoretic proof of the congruence subgroup property for $Aut(F_2)$, the group of automorphisms of a free group on two generators. This result was first proved by Asada using techniques from anabelian geometry, and our proof is, to a large extent, a translation of Asadas proof into group-theoretic language. This translation enables us to simplify many parts of Asadas original argument and prove a quantitative version of the congruence subgroup property for $Aut(F_2)$.
Let $Gamma_d(q)$ denote the group whose Cayley graph with respect to a particular generating set is the Diestel-Leader graph $DL_d(q)$, as described by Bartholdi, Neuhauser and Woess. We compute both $Aut(Gamma_d(q))$ and $Out(Gamma_d(q))$ for $d geq 2$, and apply our results to count twisted conjugacy classes in these groups when $d geq 3$. Specifically, we show that when $d geq 3$, the groups $Gamma_d(q)$ have property $R_{infty}$, that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when $d=2$ the lamplighter groups $Gamma_2(q)=L_q = {mathbb Z}_q wr {mathbb Z}$ have property $R_{infty}$ if and only if $(q,6) eq 1$.
We show that the higher rank lamplighter groups, or Diestel-Leader groups $Gamma_d(q)$ for $d geq 3$, are graph automatic. This introduces a new family of graph automatic groups which are not automatic.
We prove that the automorphism group of the braid group on four strands acts faithfully and geometrically on a CAT(0) 2-complex. This implies that the automorphism group of the free group of rank two acts faithfully and geometrically on a CAT(0) 2-complex, in contrast to the situation for rank three and above.
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