New examples of totally disconnected locally compact. G g1 n, where g1 g is the connected identity component and n a totally disconnected, stable normal subgroup i. 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. Any topological group g has a largest connected subgroup g0.
The topology of a totally disconnected locally compact t. Totally disconnected locally compact hausdorff spaces. G is locally compact if and only if there is one point of g with a local basis of compact sets. 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. Existence of compact totally disconnected groups with a. Elementary totally disconnected locally compact groups. A topological space in which any subset containing more than one point is disconnected cf. These notes have been adapted mostly from the material in the classical text mz, chapters 1 and 2, and from rv. Let xeb a loalcly ompcact hadamard space and geb a totally disconnected group acting ontinuously,c properly and ompcocactly. Universal covering spaces and fundamental groups in algebraic geometry as schemes parravi vakil etkirsten wickelgren. Siebert also characterized the connected locally compact.
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. 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. Introduction to the cohomology of topological groups igor minevich december 4, 20. Let g be a topological group, then gg0 is the totally disconnected group of connected components of g. Moskalenko 1 ukrainian mathematical journal volume 31, pages 154 158 1979 cite this article. Consider representations of gon vectorspaces over a eld kof characteristic zero. 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. A locally compact group g is totally disconnected if the identity 1 g is its own connected component. Smooth representations of totally disconnected groups.
In particular, they are considered for diffeomorphism groups and loop groups of manifolds on nonarchimedean. The seventh paper is a note on the topological group c 0 by michael megrelishvili. G is a topological group if g is a group and a topological space such that x. The topological product and the topological sum of totallydisconnected spaces, as well as any subspace of a totallydisconnected space, are totally disconnected. In mathematics, a totally disconnected group is a topological group that is totally disconnected. To to this, we shall need some more elementary results on topological groups. The purpose of the present note is to provide some partial answers to this question. If g is a topological group, however, there are many cohomology theories hng. A topological space is totally disconnected if every connected subset has at most one. These notes have been adapted mostly from the material in the classical text mz, chapters 1 and 2, and from rv, chapter 1. Chapter 1 topological groups topological groups have the algebraic structure of a group and the topologi. A topological group is said to be connected, totally disconnected, compact, locally compact, etc. The group g is a connected locally compact group, and the group of left cosets gg is a totally disconnected locally compact t. Normal subgroup structure of totally disconnected locally compact groups colin d.
A crash course in topological groups cornell university. Universal covering spaces and fundamental groups in. Structure theory of totally disconnected locally compact groups via graphs and permutations, canad j math 542002, 795827. Subgroups of a topological group such that quotient space is totally disconnected. It is well known, that g e is a normal subgroup of g, thus we may speak about quotient group. Stochastic processes on totally disconnected topological groups are investigated. We use tdlcgroup as shorthand for \totally disconnected locally compact group. The nub of an automorphism of a totally disconnected, locally compact group volume 34 issue 4 george a. Totally disconnected and locally compact heisenbergweyl. We will discuss some of these and talk about when they. However, it is easily seen that e x e and e are ams 1980 subject classifications. Locally compact, totally disconnected contraction groups. A space is totally disconnected if every connected subset of it contains at most one point. Totallydisconnected space encyclopedia of mathematics.
We may view g as an extension of g0 by a totally disconnected locally compact topological group gg0. A natural candidate for a 2dimensional totally disconnected topological group is e x e. Locally normal subgroups of totally disconnected groups. The nub of an automorphism of a totally disconnected. The group gis a connected locally compact group, and the group of left cosets ggis a totally disconnected locally compact t. The corresponding representation of the heisenbergweyl group hw. Universal covering spaces and fundamental groups in algebraic geometry 3. Featured on meta feedback on q2 2020 community roadmap. New examples of totally disconnected locally compact groups murray elder, george willis gacgta 2012, d usseldorf. This follows from the fact that the space is sequentially compact but not compact. Lets call this a group of totally disconnected tdtype.
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. Compact and totally disconnected topological group pronite group inverse limit of nite groups familiar examples. Introduction to topological groups dikran dikranjan to the memory of ivan prodanov 1935 1985. Isolating this basis requires a couple of classical results from pointset topology. The study of locally compact groups therefore in principle, although not always in practice, reduces to studying connected locally compact groups and t. In the complementary case of totally disconnected, locally compact t. The conjecture that no padic group, p a prime, can act effectively on a connected topological manifold, is namely known to be equivalent to. Pdf topological entropy in totally disconnected locally. An introduction to totally disconnected locally compact groups. Smooth representations of totally disconnected groups let gbe a locally compact, hausdor topological group with a countable basis, and totally disconnected. 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. Normal subgroup structure of totally disconnected locally. Some algebraic properties of compact topological groups.
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. Recall a space is totally disconnected if the only connected subsets are singletons onepoint subsets. We wish to show that a group gis pro nite if and only if it is a compact, hausdor, and totally disconnected topological group. These notes have been adapted mostly from the material in the. 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. Cyclic homology of totally disconnected groups acting on. Homology and topological full groups of etale groupoids on. Introduction let g be a totally disconnected group and denote by c. Take this to mean that ghas a local basis at the identity consisting of compact open subgroups. Homology and topological full groups of etale groupoids on totally disconnected spaces.
The topological full group g consists of all homeomorphisms on g0 whose graph is contained in the groupoid g as. Topological entropy in totally disconnected locally compact groups. Existence of compact totally disconnected groups with a locally compact but noncompact group of topological automorphisms z. Of special interest is the case where g is a padic group, p a prime. An equivalent condition is that the connected component of any point in the space is that point itself. Homology and topological full groups of etale groupoids on totally disconnected spaces hiroki matui graduate school of science chiba university inageku, chiba 2638522, japan. Is there a totally disconnected topological space which is not hausdorff. Theorems on compact totally disconnected semigroups and lattices1 katsumi numakura in the theory of topological algebras, it is wellknown that a compact totally disconnected group or a compact totally disconnected ring is a projective limit of finite groups or finite rings. Show that if h is a normal subgroup of g, then h is also a normal subgroup of g. Available formats pdf please select a format to send. Actions of totally disconnected groups and equivariant.
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. G is locally path connected if and only if there is one point of g with a local basis of open, path connected sets. The compact case has been heavily studied these are the profinite groups but for a long. 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.
Finiteness properties for totally disconnected locally. 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. Keywords harmonic analysis p adic numbers topological groups. Amenable groups and hadamard spaces with a totally. Introduction to the cohomology of topological groups. It is also shown that a natural homomorphism, called the index map, from the topological full group to h1 is.
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