From: Dave Rusin Subject: Re: group compactification Date: Sun, 21 Feb 1999 17:20:38 -0600 (CST) Newsgroups: [missing] To: adamp@math.uchicago.edu [Message from adamp, edited to fit messages which follow] >Consider the map G->bG from a discrete (nonabelian) group >to its group compactification. >Is it always a monomorphism? > >By a group compactification I mean the initial >homomorphism G->bG from a discrete group G to a compact group bG such that >the image of G is dense in bG. I think it is usually called the Bohr >compactification. The compactification map is not required to be a >homeomorphism onto its image. Existence and uniqueness, up to isomorphism, >are obvious. I would like to know if it is a monomorphism. If G is not >discrete I think there are examples when this compactification has a >kernel. I don't know if this is the case when G is nonabelian discrete. In >the abelian case G can be seen as a module over integers Z. Every cyclic >Z-module embeds into a circle which is compact and injective as a Z-module >hence the Bohr compactification can not have a kernel in this case. > >Thanks, >Adam Przezdziecki ============================================================================== From: Dave Rusin Subject: Re: group compactification Date: Sat, 20 Feb 1999 23:45:07 -0600 (CST) Newsgroups: [missing] To: adamp@math.uchicago.edu Forgive me for being dense, but what is bG? The compactifications of general topological spaces I have seen -- one-point cpt., Stone-Cech cpt. --- come equipped with 1-to-1 maps from X to X*; while it's not clear to me which compactifications can be given a group structure, and while the map from X to X* need not be a homeomorphism onto its image, in general, it nonetheless does seem difficult to envision a good notion of "compactification" which would have a kernel! dave ============================================================================== From: Adam Przezdziecki Subject: Re: group compactification Date: Sun, 21 Feb 1999 16:13:54 -0600 (CST) Newsgroups: [missing] To: rusin@math.niu.edu (Dave Rusin) Sorry for being unclear. By a group compactification I meant the initial homomorphism G->bG from a discrete group G to a compact group bG such that the image of G is dense in bG. I think it is usually called the Bohr compactification. The compactification map is not required to be a homeomorphism onto its image. Existence an uniqueness, up to isomorphism, are obvious. I would like to know if it is a monomorphism. If G is not discrete I think there are examples when this compactification has a kernel. I don't know if this is the case when G is nonabelian discrete. In the abelian case G can be seen as a module over integers Z. Every cyclic Z-module embeds into a circle which is compact and injective as a Z-module hence the Bohr compactification can not have a kernel in this case. If bG is the one-point compactification of G then G embeds in bG but the action of G on bG fixes the point at "infinity" hence the group structure does not extend to bG. If bG is the Stone-Cech compactification then the isolated points of bG are precisely those that came from G hence bG is not homogeneous (homeomorphisms don't act transitively on bG) so bG again does not admit any group structure compatible with the topology. Adam [previous letter quoted -- djr] ============================================================================== From: Torsten Ekedahl Subject: Re: group compactification Date: 22 Feb 1999 12:30:03 -0600 Newsgroups: sci.math.research Adam Przezdziecki writes: > Consider the map G->bG from a discrete (nonabelian) group > to its group compactification. > Is it always a monomorphism? A compact group is "residually linear" in the sense that any non-identity element can be mapped to a non-identity element under a homomorphism to some GL_n(C). (This follows from the Peter-Weyl theorem.) Hence any group that can be embedded into a compact group is residually linear. There are lots of non-residually linear groups. For instance any finitely generated residually linear group is residually finite and there are lots of f.g. non-residually finite groups. ============================================================================== From: Joerg Winkelmann Subject: Re: group compactification Date: 22 Feb 1999 12:30:10 -0600 Newsgroups: sci.math.research Adam Przezdziecki wrote: > Consider the map G->bG from a discrete (nonabelian) group > to its group compactification. > Is it always a monomorphism? > By a group compactification I mean the initial > homomorphism G->bG from a discrete group G to a compact group bG such that > the image of G is dense in bG. I think it is usually called the Bohr > compactification. The compactification map is not required to be a > homeomorphism onto its image. Existence and uniqueness, up to isomorphism, > are obvious. I would like to know if it is a monomorphism. If G is not > discrete I think there are examples when this compactification has a > kernel. I don't know if this is the case when G is nonabelian discrete. In > the abelian case G can be seen as a module over integers Z. Every cyclic > Z-module embeds into a circle which is compact and injective as a Z-module > hence the Bohr compactification can not have a kernel in this case. Every compact group has a faithful linear representation on a Hilbert space, namely the space of L^2 functions on this group and the union of invariant finite-dimensional subspaces in this Hilbert space is dense. Using this, it follows that given a compact group K and an element g in K there is a finite dimensional representation f of K such that f(g) is not trivial. (f depends in general on g, unless K is assumed to be a Lie group.) Now let G be a finitely generated discrete group. If G can be embedded into a compact group K, then for every g in G there exists a finite dimensional representation f with f(g) non-trivial. However, there are countable groups such that the intersection of the kernels of all finite dimensional representations is non-trivial. Thus for these groups the Bohr compactification can not be injective. AFAIK examples of such finitely generated groups can be found using lattices in universal coverings of real semisimple algebraic groups. Maybe this is contained in: Raghunathan, M. S.(F-IHES) Torsion in cocompact lattices in coverings of ${\rm Spin}(2,\,n)$. Math. Ann. 266 (1984), no. 4, 403--419. HTH Joerg Winkelmann -- jwinkel@member.ams.org Mathematisches Institut der Universitaet Basel http://www.cplx.ruhr-uni-bochum.de/~jw/ ============================================================================== From: flor Subject: Re: group compactification Date: Tue, 23 Feb 1999 20:27:57 Newsgroups: sci.math.research The original question refers to a situation which was treated long ago. A group admitting an injective homomorphism into some compact group is called "maximally almost periodic". The original question thus amounts to the following problem: Is every discrete group maximally almost periodic? Questions of this kind were studied by J. von Neumann and E. Wigner in their 1940 paper "Minimally almost periodic groups". The simplest example of a group which is not maximally almost periodic may well be the following: let G be the group of affine selfmaps of the real line, under composition. (This is a two-dimensional, noncommutative Lie group.) It is easily seen that in this group, all nontrivial translations form just one conjugacy class. Any homomorphism of G into a compact group thus maps all translations into a single conjugacy class; call it C. Every conjugacy class in a compact group is closed; so C is closed. Now let c be the image of any nontrivial translation. Not only c is an element of C but every power c^n, n/=0, also is in C. In every compact group, the closure of the set of all positive powers of any element contains the unit element. Hence, our conjugacy class C contains the unit element which, of course, is conjugate only to itself. So C ={e}; that ist, all translations are mapped into e, under every homomorphism of G to a compact group. This proof did not assume that the homomorphism should be continuous; hence it works for G with the discrete topology.