Minimally almost periodic totally disconnected groups

In this paper we prove that every closed noncompact group of isometries of a homogeneous tree which acts transitively on the tree boundary contains a normal closed cocompact subgroup which is minimally almost periodic. Moreover we prove that is a topologically simple group.

     

Rearrangements of functions on totally disconnected groups

A result of B.B. Wells Jr. claims that every complex valued continuous function on the compact ring of p-adic integers has a rearrangement which belongs to a certain class (W) of functions having absolutely convergent Fourier series. We point out that this is not the case since every real valued function from (W) has Lebesgue null range. On the other hand we prove the existence of a rearrangement with absolutely convergent Fourier series for every continuous real valued function on and on some other compact metric totally disconnected Abelian groups. We leave open if the same holds for all continuous complex valued functions on .     

Chern character for totally disconnected groups

In this paper we construct a bivariant Chern character for the equivariant KK-theory of a totally disconnected group with values in bivariant equivariant cohomology in the sense of Baum and Schneider. We prove in particular that the complexified left hand side of the Baum–Connes conjecture for a totally disconnected group is isomorphic to cosheaf homology. Moreover, it is shown that our transformation extends the Chern character defined by Baum and Schneider for profinite groups.     

Bredon-Illman cohomology with actions of totally disconnected groups

We study the Bredon-Illman cohomology with local coefficients for a G-space X in the case of G being a totally disconnected, locally compact group. We prove that any short exact sequence of equivariant local coefficients systems on X gives a long exact sequence of the associated Bredon-Illman cohomology groups with local coefficients.     

Classical solutions to differential inclusions with totally disconnected right-hand side

Let F=F(t,x) be a bounded, Hausdorff continuous multifunction with compact, totally disconnected values. Given any y₀∈F(t₀,x₀), we show that the differential inclusion has a globally defined classical solution, with x(t₀)=x₀, .     

A zero-one law for random subgroups of some totally disconnected groups

Let A be a locally compact group topologically generated by d elements and let k > d. Consider the action, by precomposition, of Γ = Aut(F _k ) on the set of marked, k-generated, dense subgroups $ {D_{k,A}}: = \left\{ {\eta \in {\text{Hom}}\left( {{F_k},A} \right)\left| {\overline {\left\langle {\phi \left( {{F_k}} \right)} \right\rangle } = A} \right.} \right\} Let A be a locally compact group topologically generated by d elements and let k > d. Consider the action, by precomposition, of Γ = Aut(F _k ) on the set of marked, k-generated, dense subgroups D_k,A: = { h ? \textHom( F_k,A )| [`( á f( F_k ) ñ )] = A } {D_{k,A}}: = \left\{ {\eta \in {\text{Hom}}\left( {{F_k},A} \right)\left| {\overline {\left\langle {\phi \left( {{F_k}} \right)} \right\rangle } = A} \right.} \right\} . We prove the ergodicity of this action for the following two families of simple, totally disconnected, locally compact groups:

•	A = PSL2(K) where K is a non-Archimedean local field (of characteristic ≠ 2);
•	A = Aut0(T q+1)—the group of orientation-preserving automorphisms of a q + 1 regular tree, for q \geqslant 2.q \geqslant 2.

In contrast, a recent result of Minsky’s shows that the same action fails to be ergodic for A = PSL₂(C) and, when k is even, also for A = PSL₂(R). Therefore, if k \geqslant 4 k \geqslant 4 is even and K is a local field (with char(K) ≠ 2), the action of Aut(F _k ) on D_{k,\textPS\textL2(K)} {D_{k,{\text{PS}}{{\text{L}}_2}(K)}} is ergodic if and only if K is non-Archimedean. Ergodicity implies that every “measurable property” either holds or fails to hold for almost every k-generated dense subgroup of A.     

Absolute convergence of Fourier series on totally disconnected groups

LetG denote a totally disconnected locally compact metric abelian group with translation invariant metricd and character groupΓ _G . The Lipschitz spaces are defined by $$Lip\left( {\alpha ;p} \right) = \left\{ {f \in L^p \left( G \right):\left\| {\tau _a f - f} \right\|_p = O\left( {d\left( {a,0} \right)^\alpha } \right),a \to 0} \right\},$$ whereτ _a f:x→f(x-a) and α∈(0,1). For a suitable choice of metric it is shown that Lip (α;p)?L ^r (Γ _G ), where α>1/p+1/r?1≧0 and 1≦p≦2. In the caseG is compact the corresponding result holds for α>1/r?1/2 andp>2. In addition forG non-discrete the above result is shown to be sharp, in the sense that the range of values of α cannot be extended. The results include classical theorems of S. N. Bernstein, O. Szász and E. C. Titchmarsh.     

Actions of totally disconnected groups and equivariant singular homology

We study equivariant singular homology in the case of actions of totally disconnected locally compact groups on topological spaces. Theorem A says that if G is a totally disconnected locally compact group and X is a G-space, then any short exact sequence of covariant coefficient systems for G induces a long exact sequence of corresponding equivariant singular homology groups of the G-space X. In particular we consider the case where G is a totally disconnected compact group, i.e., a profinite group, and G acts freely on X. Of special interest is the case where G is a p-adic group, p a prime. The conjecture that no p-adic group, p a prime, can act effectively on a connected topological manifold, is namely known to be equivalent to the famous Hilbert-Smith conjecture. The Hilbert-Smith conjecture is the statement that, if a locally compact group G acts effectively on a connected topological manifold M, then G is a Lie group.     

On unique range sets for meromorphic or entire functions

This paper has studied the Gross uniqueness question ignoring multiplicity and relating multiple values, and extended and improved on some related results in [1–8]. Supported in part by National Natural Science Foundation of China     

On contraction groups of automorphisms of totally disconnected locally compact groups

We prove that recent results of Baumgartner and Willis on contraction groups of automorphisms of metrizable totally disconnected locally compact groups (Israel J. Math. 142 (2004), 221–248) remain true for non-metrizable groups. Supported by an NSERC Grant.     

Unbounded functions with almost periodic differences

The structure of continuous real-valued functions F(t) on the real line, such that for any fixed y the difference F(t+y)–F(t) is an almost periodic Bohr function, is investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 10, pp. 1409–1413, October, 1991.     

Contraction groups and scales of automorphisms of totally disconnected locally compact groups

We study contraction groups for automorphisms of totally disconnected locally compact groups using the scale of the automorphism as a tool. The contraction group is shown to be unbounded when the inverse automorphism has non-trivial scale and this scale is shown to be the inverse value of the modular function on the closure of the contraction group at the automorphism. The closure of the contraction group is represented as acting on a homogenous tree and closed contraction groups are characterised. Research supported by DFG Grant BA 1966/1–1 and ARC Grant A69700321.