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The action of a group $G$ on a set $X$ is called oligomorphic if the diagonal action on $X^n$ has finitely many orbits for each $n$. Question: Is there an infinite (maybe even finitely generated) …

The commutator subgroup of the free group $\langle a,b \rangle$ is freely generated by the set $$\lbrace [a^n,b^m] \mid n,m \in \mathbb Z, nm \neq 0 \rbrace.$$

Since you are interested in positive results (rather than counterexamples) in the case when the pro-finite completions of two groups agree, let me mention the following result from Martin R. Bridson …

If some element centralizes the complex conjugation, then it must preserve the real numbers as a set. Now, since any automorphism of the real numbers preserves the set of squares, it must preserve the …

The answer is: This does not hold in general. If the group in question is finitely generated, then the maps into the colimit will eventually be surjective. If the group in question is also finitely p …

I think that this is known in the operator algebra commmunity, but it is also a consequence of the proof in the sofic case, which was obtained in Elek, Gábor, Szabó, Endre, On sofic groups. J. Group …

As Mark Sapir pointed out, this is of course false. There are tons of infinite dimensional irreducible representations of residually finite groups. In fact, I think that any finitely generated group f …

This is just an idea, not a solution. In Marc Burger, Shahar Mozes, Lattices in product of trees, Inst. Hautes Etudes Sci. Publ. Math. 2000 (92) pp. 151-194 the authors study a class of groups G whi …

This concerns the updated question. The answer is yes. If $G$ is represented as the union of a finite number of subsets $A_1, \dots,A_n$, then one of the subsets generates a finite index subgroup of …

I do not know whether this answers you question, but such a group is an extension of $K/K \cap L \times K / K \cap L$ by $K \cap L$, i.e. $$1 \to K \cap L \to KL \to K/K \cap L \times L/K \cap L \to …

There are always derivations of the form $\delta_b(a) = ab - b\varepsilon(a)$ for $b \in R[G]$, which are called inner derivations. Derivations modulo inner derivations are classified by the first gro …

Related to a question by Mark Sapir (see here) and a question by Kate Juschenko (see here), let me ask the following: Question: Let $G$ be a finitely generated group and let $\varepsilon>0$. Is th …

Let $G$ be a group with two generators. Suppose that all non-trivial words of length less or equal $n$ in the generators and their inverses define non-trivial elements in $G$. Question: How many o …

My questions are: Is there any group, which cannot be written as the quotient of a residually finite group by an amenable normal subgroup? Is it possible for large classes of groups? and Is …

Let $G$ be a non-trivial injective group and $g \in G$ non-trivial. From category theory, $G \times G$ is injective as well. But, now $G \times G$ embeds into a group $H$, where $(g,e)$ and $(e,g)$ ar …