# Questions tagged [topological-groups]

A topological group is a group $G$ together with a topology on the elements of $G$ such that the group operation and group inverse function are both continuous (with respect to the topology).

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### Finite covolume of uniform lattice in quotient group

Let $G$ be a locally compact group, let $N \leq G$ be a (proper) closed normal subgroup and let $\Gamma \leq G$ be a uniform lattice, i.e., a discrete subgroup such that $G/\Gamma$ Is compact. Suppose ...
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### Restrictions on pointed lifts of isometries

Let $M$ be a closed Riemannian manifold and let $f$ be an isometry of $M$ that fixes a point $\ast \in M$ and acts trivially on $\Gamma := \pi_1(M,\ast)$. Then there is a unique isometry $\tilde{f}$ ...
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### Trying to understand “a refinement of the Peter–Weyl theorem” by Lusztig

"A refinement of the Peter–Weyl theorem" is the title of Chapter 29 in Lusztig's "Introduction to quantum groups" (Birkhäuser 2010, reprint of the 1994 edition). This chapter is ...
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### Morphism of R-algebras between R-adic algebras

Let $R$ be a $f$-adic ring. Let $A$ and $B$ be $R$-adic algebras. I would like to show that any morphism of $R$-algebras between $A$ and $B$ is actually adic.
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### Haar measure coming from Pontryagin duality v/s Fourier inversion

Not research but advertising this question from mse in case someone wants to answer. I'm struggling with some bookkeeping associated with the Pontryagin duality theorem. I'm thinking about the first ...
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### Metric completion of an algebraically closed field is algebraically closed?

Let $F$ be a complete metric topological field. Suppose there is a subfield $F_1 \subset F$, algebraically closed and topoolgically dense in $F$. Must $F$ itself be algebraically closed? We can ...
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### About locally compact groups without compact subgroups

Is every Hausdorff, locally compact group that does not contain any non-trivial compact group, finitely dimensional?
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### Are locally compact, Hausdorff, locally path-connected topological groups locally Euclidean?

Is every locally compact, Hausdorff, locally path-connected topological group $G$ locally Euclidean? (That would imply of course also being a Lie group.) Is it true when countable basis is assumed? I ...
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### With a linear representation, how does the continuity of $G \to \mathrm{GL}(V)$ relate to that of $G \times V \to V$?

I'm currently reading Traces of Hecke Operators by Knightly and Li, while simultaneously revisiting the adelic/representation-theoretic point of view on automorphic forms. In Knightly and Li, they ...
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### Algebraic rigidity in the automorphism group of the Cantor set

Let C be a Cantor set (middle third). Now we know that C is a totally disconnected compact topological space with the natural topology (i.e., $C=\{0,1\}^{\mathbb{N}}$). Let G:=Homeo(C) be the set of ...
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### What is the smallest number of nowhere dense affine subsets covering a topological group?

$\DeclareMathOperator\cov{cov}\newcommand\A{\text A}$A subset $A$ of a group $G$ is called affine if $A=xHy$ for some subgroup $H\subseteq G$ and some $x,y\in G$. Given a non-discrete topological ...
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