Of special interest is the case where g is a padic group, p a prime. The study of locally compact groups therefore in principle, although not always in practice, reduces to studying connected locally compact groups and t. Reid abstract the present article is a summary of joint work of the author and phillip wesolek on the normal subgroup structure of totally disconnected locally compact secondcountable t. A topological group is said to be connected, totally disconnected, compact, locally compact, etc. G g1 n, where g1 g is the connected identity component and n a totally disconnected, stable normal subgroup i. Moskalenko 1 ukrainian mathematical journal volume 31, pages 154 158 1979 cite this article. Existence of compact totally disconnected groups with a locally compact but noncompact group of topological automorphisms z. The topology of a totally disconnected locally compact t. Recall a space is totally disconnected if the only connected subsets are singletons onepoint subsets. A crash course in topological groups cornell university. Take this to mean that ghas a local basis at the identity consisting of compact open subgroups. Compact and totally disconnected topological group pronite group inverse limit of nite groups familiar examples. Smooth representations of totally disconnected groups. The group g is a connected locally compact group, and the group of left cosets gg is a totally disconnected locally compact t.
We wish to show that a group gis pro nite if and only if it is a compact, hausdor, and totally disconnected topological group. Consider representations of gon vectorspaces over a eld kof characteristic zero. Smooth representations of totally disconnected groups let gbe a locally compact, hausdor topological group with a countable basis, and totally disconnected. Pdf topological entropy in totally disconnected locally. This follows from the fact that the space is sequentially compact but not compact. It is well known, that g e is a normal subgroup of g, thus we may speak about quotient group. Keywords harmonic analysis p adic numbers topological groups. Topological entropy in totally disconnected locally compact groups article pdf available in ergodic theory and dynamical systems 1 july 2015 with 57 reads how we measure reads. Show that if h is an abelian subgroup of g, then h is also an abelian subgroup of g. Normal subgroup structure of totally disconnected locally. New examples of totally disconnected locally compact groups murray elder, george willis gacgta 2012, d usseldorf.
The compact case has been heavily studied these are the profinite groups but for a long. Stochastic processes on totally disconnected topological groups are investigated. However, it is easily seen that e x e and e are ams 1980 subject classifications. On the other hand, we have a notion of a locally profinite group, a hausdorff topological group which has a neighborhood basis of the identity consisting of open compact subgroups. We will discuss some of these and talk about when they. In the complementary case of totally disconnected, locally compact t. The purpose of the present note is to provide some partial answers to this question. We may view g as an extension of g0 by a totally disconnected locally compact topological group gg0. Homology and topological full groups of etale groupoids on totally disconnected spaces. Finiteness properties for totally disconnected locally. A topological space is totally disconnected if every connected subset has at most one. Amenable groups and hadamard spaces with a totally.
An equivalent condition is that the connected component of any point in the space is that point itself. Any topological group g has a largest connected subgroup g0. To to this, we shall need some more elementary results on topological groups. It is also shown that a natural homomorphism, called the index map, from the topological full group to h1 is. Universal covering spaces and fundamental groups in algebraic geometry as schemes parravi vakil etkirsten wickelgren. Homology and topological full groups of etale groupoids on. Existence of compact totally disconnected groups with a. The topological full group g consists of all homeomorphisms on g0 whose graph is contained in the groupoid g as. Some algebraic properties of compact topological groups. Chapter 1 topological groups topological groups have the algebraic structure of a group and the topologi.
A topological space in which any subset containing more than one point is disconnected cf. Let g be a topological group, then gg0 is the totally disconnected group of connected components of g. Lets call this a group of totally disconnected tdtype. Show that if h is a normal subgroup of g, then h is also a normal subgroup of g.
We use tdlcgroup as shorthand for \totally disconnected locally compact group. Theorem van dantzig, 1932 a topological group g is a tdlcgroup if, and only if, g has a neighbourhood basis at 1 g consisting of compact open subgroups. Topological groups math 519 the purpose of these notes is to give a mostly selfcontained topological background for the study of the representations of locally compact totally disconnected groups, as in bz or b, chapter 4. A space is totally disconnected if every connected subset of it contains at most one point. An introduction to totally disconnected locally compact groups. Totally disconnected locally compact groups with a linear. Universal covering spaces and fundamental groups in algebraic geometry 3. Actions of totally disconnected groups and equivariant. The seventh paper is a note on the topological group c 0 by michael megrelishvili.
A separable totally disconnected ordered topological group is proved to be nonarchimedean metrizable while the converse is shown to be false by means of an example. G is locally compact if and only if there is one point of g with a local basis of compact sets. The topological product and the topological sum of totallydisconnected spaces, as well as any subspace of a totallydisconnected space, are totally disconnected. Locally compact, totally disconnected contraction groups. Introduction let g be a totally disconnected group and denote by c. G is locally path connected if and only if there is one point of g with a local basis of open, path connected sets. Isolating this basis requires a couple of classical results from pointset topology. Siebert also characterized the connected locally compact. Totallydisconnected space encyclopedia of mathematics. Elementary totally disconnected locally compact groups.
The group gis a connected locally compact group, and the group of left cosets ggis a totally disconnected locally compact t. Subgroups of a topological group such that quotient space is totally disconnected. Homology and topological full groups of etale groupoids on totally disconnected spaces hiroki matui graduate school of science chiba university inageku, chiba 2638522, japan. Introduction to topological groups dikran dikranjan to the memory of ivan prodanov abstract these notes provide a brief introduction to topological groups with a special emphasis on pontryaginvan kampens duality theorem for locally compact abelian groups. If g is a topological group, however, there are many cohomology theories hng. Available formats pdf please select a format to send. Is there a totally disconnected topological space which is not hausdorff. Among the applications, we show that if a nondiscrete, compactly generated, topologically simple, totally disconnected locally compact group is locally linear, then it is a simple algebraic group over a local. Normal subgroup structure of totally disconnected locally compact groups colin d. Introduction to the cohomology of topological groups igor minevich december 4, 20. Let xeb a loalcly ompcact hadamard space and geb a totally disconnected group acting ontinuously,c properly and ompcocactly. A necessary and sufficient condition for a topological group whose topology can be induced by a total order compatible with the group structure is given and such groups are called ordered or orderable topological groups. Totally disconnected and locally compact heisenbergweyl.
Introduction to the cohomology of topological groups. The nub of an automorphism of a totally disconnected. Structure theory of totally disconnected locally compact groups via graphs and permutations, canad j math 542002, 795827. In particular, they are considered for diffeomorphism groups and loop groups of manifolds on nonarchimedean. A locally compact group g is totally disconnected if the identity 1 g is its own connected component. Background new tools applications to simple groups totally disconnected, locally compact groups simple groups topological groups a topological group is a group that is also a topological space, such that x. A natural candidate for a 2dimensional totally disconnected topological group is e x e. G is a topological group if g is a group and a topological space such that x. Featured on meta feedback on q2 2020 community roadmap. Cyclic homology of totally disconnected groups acting on. New examples of totally disconnected locally compact. Universal covering spaces and fundamental groups in. The corresponding representation of the heisenbergweyl group hw.
Totally disconnected locally compact hausdorff spaces. We prove that every separable metrizable totally disconnected topological group is topologically isomorphic to the autohomeomorphism group of some separable metrizable space, when given the. These notes have been adapted mostly from the material in the. Introduction to topological groups dikran dikranjan to the memory of ivan prodanov 1935 1985. The conjecture that no padic group, p a prime, can act effectively on a connected topological manifold, is namely known to be equivalent to. Such topological groups are necessarily hausdorff interest centres on locally compact totally disconnected groups variously referred to as groups of tdtype, locally profinite groups, t. These notes have been adapted mostly from the material in the classical text mz, chapters 1 and 2, and from rv, chapter 1.
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