The following definition has arisen naturally in two papers of mine. The papers are on rather unrelated topics; of course they are within my narrow interests, so there's some symbolic dynamics connection, but really in both cases I just needed to find good "generic elements / finite subsets" of the group, and I can't help but wonder if this has appeared elsewhere, or is even something I know but am just not recognizing.

Definition. A topological group $G$ is

splendidif for all compact $C \subset G$, there exists $g \in G$ such that $gCg \cap C = \emptyset$.

I'll mention some sufficient conditions for being splendid and give some examples. First, abelian groups are usually splendid unless obviously not splendid, and this passes to preimages, as follows:

Lemma. Let $G$ be a topological group that has a continuous abelian (surjective) quotient $(H,+)$ such that $\{2h \;|\; h \in H\} \leq H$ is non-compact. Then $G$ is splendid.

For simplicity let's now concentrate on discrete infinite groups, so compact means finite. The previous lemma shows that e.g. free abelian groups, free groups and Thompson's $F$ are splendid, since they have positive rank abelianizations. Every finitely-generated infinite discrete left-orderable amenable group is indicable by a result of Morris, thus splendid by the previous lemma. (You can generalize easily to locally indicable, thus all discrete left-orderable amenable groups.)

Not every group is splendid: the infinite dihedral group $\langle a, b \;|\; a^2 = b^2 = e \rangle$ with the discrete topology is not splendid, namely pick $C = \{e, a\}$. Besides the above lemma, a way to show splendedness is that every non-splendid group that has an infinite abelian subgroup is at least a bit little dihedral (I worked this out just for this post, so take with a grain of salt):

Lemma. Suppose $G$ is a non-splendid discrete infinite group. If $H \leq G$ is abelian, then there is a finite-index subgroup $H' \leq H$ and some $a \in G$ such that $h^a = h^{-1}$ for all $h \in H'$.

Proof: Let $C$ be the finite set proving non-splendidness. Then $hCh = C \neq \emptyset$ for all $h \in H$. Since $H$ is abelian, $H$ acts on $G$ from the left by $h * g = hgh$. Since $h*C \cap C \neq \emptyset$, writing $S_a \leq H$ for the stabilizer of $a \in C$ and $T_{a,b}$ for the transporter $\{h \in H \;|\; ha = b\}$, we have $$ H = \bigcup_{a, b \in C} T_{a, b} = \bigcup_{a, b \in C \\ T_{a,b} \neq \emptyset} h_{a, b} S_a $$ for some choices $h_{a,b} \in H$. By a result of Neumann, some $H' = S_a$ has finite index in $H$, and for $h \in H'$ we have $hah = a \implies h^{-1} = aha^{-1}$. Square.

In particular a group is splendid if there is an element of infinite order such that no power of it is conjugate to its inverse. For example this lemma applies to Thompson's $T$ (which has no nontrivial abelian quotients so the first lemma does not apply): Pick an element $f$ that fixes a unique interval $I \subset S^1$ and has derivative $>1$ after the right endpoint of $I$. Then $\langle f \rangle$ is of infinite order and every positive power $f^i$ has the same property as $f$, while no negative power $f^{-1}$ does. This property is preserved under conjugacy, so no $f^i$ is conjugate to its inverse.

Question. Does this have a more standard, even if possibly less splendid, existing name? Alternative characterizations that are easier to verify?

You can refine the definition as follows, for discrete groups: say $G$ is **$k$-splendid** if the splendidness condition holds for all $|C| \leq k$. Every infinite group is $1$-splendid, while the dihedral group is not $2$-splendid. A characterization of $k$-splendidness for small $k$ would also be of interest.

A list of my favorite groups and whether or not they are splendid is also of interest to me, I tried to compile one, but after working out those Thompson's's I got stuck on the Grigorchuk group.

*Neumann, B. H.*, Groups covered by finitely many cosets, Publ. Math., Debrecen 3, 227-242 (1954). ZBL0057.25603.

*Morris, Dave Witte*, **Amenable groups that act on the line.**, Algebr. Geom. Topol. 6, 2509-2518 (2006). ZBL1185.20042.

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