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Say a topological group $G$ is topologically characteristically simple if there does not exist a closed subgroup $1 < K < G$ such that $K$ is invariant under all automorphisms of $G$ (here `automorphi …

I managed to answer my own question a few years later, here: https://www.degruyter.com/document/doi/10.1515/jgth-2020-0107/html C. Reid, A classification of the abelian minimal closed normal subgroups …

I'm looking at a certain class of groups $G$ that come with a partial order $\le$ on the elements. So far it looks like $(G,\le)$ has the following properties: The partial order is invariant under …

Let $C = \mathbb{Z}/d\mathbb{Z}$ ($d \ge 0$). Let $D = \langle a_c : c \in C, t \mid a^2_c = t^d = 1, ta_ct^{-1} = a_{c+1} \rangle$. let $E$ be the subgroup generated by $\{a_c : c \in C\}$ and let …

As YCor says, this is a community wiki 'big list'-type question. Here are a couple of examples I've heard of: One context where fairly exotic-looking groups arise from dynamical considerations is i …

OK, here is an attempted answer under the assumption that $G$ is locally compact, which can perhaps be refined to give a general answer for Polish groups. A good reference would still be appreciated …

Let $G$ and $H$ be Polish groups and let $\psi: G \rightarrow H$ be a continuous injective homomorphism such that $\psi(G)$ is normal in $H$. Then $H$ acts on $G$ by conjugation via $\psi$, in other …

EDIT: Here is a more specific question. Let $G$ be a compact group and let $w$ be a word in $d$ variables. Then the solution set $S$ of the equation of $w=1$ is a closed subset of the product $G^d$ …

For which properties $(P)$ is the following statement known to be true? In any locally compact group $G$, the elements of $G$ that satisfy $(P)$ form a closed subset of $G$. In other words, the limi …

This question has already been answered, but I thought it would be a good idea to draw attention to some things that have appeared since the question was asked: There is an upcoming book by Yves de …

It's quite a specific family of examples, but you might find this paper of interest as an example of how a compactly generated simple t.d.l.c. group can fail to have any lattices: Bader, Caprace, Gel …

A pro-$p$ group of finite subgroup rank has an open subgroup $P$ that is uniformly powerful, meaning that $[P,P]$ is contained in the group generated by $2p$-th powers in $P$, and raising elements to …

Let $G$ be a Hausdorff locally compact group and let $\alpha$ be an automorphism of $G$. Say $\alpha$ is (forwards) topologically recurrent if for all $g \in G$ and all neighbourhoods $O$ of $g$, the …

Let $M$ be a meet-semilattice with a least element $0$. Suppose there is an order-reversing involution $a \mapsto -a$ on $M$ such that for all $a, b \in M$, $a \wedge b = 0$ if and only if $b \le -a$ …

Let $G$ be a topological group, and let $M$ be the intersection of all conjugacy-invariant neighbourhoods of the identity in $G$ (in other words, the set of elements that can be taken arbitarily close …