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Register a new userOn small abstract quotients of Lie groups and locally compact groups
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Authors
Linus Kramer
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We prove continuity results for abstract epimorphisms of locally compact groups onto finitely generated groups.
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We are concerned with questions of the following type. Suppose that $G$ and $K$ are topological groups belonging to a certain class $cal K$ of spaces, and suppose that $phi:K to G$ is an abstract (i.e. not necessarily continuous) surjective group homomorphism. Under what conditions on the group $G$ and the kernel is the homomorphism $phi$ automatically continuous and open? Questions of this type have a long history and were studied in particular for the case that $G$ and $K$ are Lie groups, compact groups, or Polish groups. We develop an axiomatic approach, which allows us to resolve the question uniformly for different classes of topological groups. In this way we are able to extend the classical results about automatic continuity to a much more general setting.
Suppose that $X=G/K$ is the quotient of a locally compact group by a closed subgroup. If $X$ is locally contractible and connected, we prove that $X$ is a manifold. If the $G$-action is faithful, then $G$ is a Lie group.
We show that every abstract homomorphism $varphi$ from a locally compact group $L$ to a graph product $G_Gamma$, endowed with the discrete topology, is either continuous or $varphi(L)$ lies in a small parabolic subgroup. In particular, every locally compact group topology on a graph product whose graph is not small is discrete. This extends earlier work by Morris-Nickolas. We also show the following. If $L$ is a locally compact group and if $G$ is a discrete group which contains no infinite torsion group and no infinitely generated abelian group, then every abstract homomorphism $varphi:Lto G$ is either continuous, or $varphi(L)$ is contained in the normalizer of a finite nontrivial subgroup of $G$. As an application we obtain results concerning the continuity of homomorphisms from locally compact groups to Artin and Coxeter groups.
We study abstract group actions of locally compact Hausdorff groups on CAT(0) spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris-Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion free CAT(0) group is continuous.
We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.
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