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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 …

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 …

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 …

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 …

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 …

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 $G^d$ …

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 $M$ 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 $b \le -a$ …

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 …

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 …

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 …

In group theory, it's often very useful to know whether a family of subgroups (eg normal subgroups, Zariski-closed subgroups, ...) satisfies an ascending chain condition or a descending chain conditio …

Does anyone know if there has been much work done on radical and semisimple classes in the sense of Kurosh within the category of topological groups (or subcategories thereof)? For instance, for a ra …

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 …