Questions

Below I give a list of still open questions I have previously asked in my papers. Motivation and background for these questions can be found by following the source links. I am interested in all these questions and the amount I am interested in them is inconsistant and hard to quantify. However, as I enjoy ranking things, I have attempted to list them in order of how much I want to know the answer (most first). After the list I also give the questions I’ve asked which have since been ansered. If you know the answers to any of the unanswered questions then please let me know!

Last updated 03/06/2026

Still Open Questions

1. (source) What is the commutative subgroup width of $\text{Sym}(\mathbb{N})$?

2. (source) Is it true that there is a semigroup compatible with precisely $\kappa$ Polish semigroup topologies if and only if $\kappa ∈ ${$0, 1, \aleph_0, 2^{\aleph_0}, 2^{2^{\aleph_0}}$}?

3. (source) Do any of the monoids $\text{Aut}(\mathbb{Q}, ≤)$, $\text{Aut}(R)$, $\text{End}(\mathbb{Q}, ≤)$, $\text{End}(R)$ have finite commutative submonoid width?

4. (source) Are there any more Hausdorff group topologies on any of the Thompson groups F, T and V other than the pointwise topologies (with respect to one of the orbits of their usual action), the compact-open topology, joins of these, and the discrete topology?

5. (source) If G is a Polish group and G has the small index property, then does G have automatic continuity with respect to the class of second countable groups? More info can be found here.

6. (source) Is the Markov topology on the homeomorphism group of the Cantor space equal to the Zariski topology?

7. (source) Is the Markov topology on homeomorphism group of the Cantor space Hausdorff?

8. (source) Do either of the Polish topological semigroups $(\text{End}(\mathbb{Z}, ≤), T_0)$, $(\text{End}(\mathbb{N}, ≤), T_{max})$ have automatic continuity with respect to the class of second-countable topological semigroups?

9. (source) How many Hausdorff group topologies does V admit?

10. (source) Can one classify which “sufficiently large” groups of homeomorphisms of the Cantor space have a Hausdorff Zariski topology (for some reasonable notion of sufficiently large)?

11. (source) Can the category UDAFG be used to build “small” generating sets for the groups $Out(G_{n,n−1})$ $(n ≥ 2)$? In particular is the group $Out(V)$ finitely generated?

12. (source) For which space-group pairs $(X, H)$ is it the case that if $G ≤ \text{Homeo}(X)$ is finitely generated, simple and vigorous with respect to H then G is two-generated?

13. (source) Does there exist an epiRegular group which does not admit a rational cross section?

14. (source) Let $C_1$ and $C_2$ be distinct language classes in the Chomsky hierarchy such that $C_1$ is a subclass of $C_2$. Does there exist a group in epi$C_2$ but not epi$C_1$?

15. (source) Is Grigorchuk’s group epiRegular?

16. (source) Is there a finitely generated group with intermediate automorphic growth?

17. (source) Is there a finitely generated group with exponential conjugacy growth but polynomial automorphic growth?

18. (source) Is there a topological semigroup which has property $\mathbb{XX}$ with respect a non-empty subset $A$ but not with respect to its top $\mathscr{J} -class$?

19. (source) Is every finitely generated simple vigorous group a quotient of the free product of $C_2$ and $C_3$?

20. (source) Are there any groups with polynomially bounded, but not polynomial automorphic growth rates?

21. (source) Does there exist a finitely presented simple group that is not 2-generated?

22. (source) The endomorphism monoids of which homogeneous graphs have a unique Polish topology?

23. (source) Which diagram groups have decidable Diophantine problem?

24. (source) Can automorphic growth be fully classified for all virtually abelian groups in a similar manner to Theorem 4.18?

25. (source) Is the single equation problem decidable in Thompson’s group F?

26. (source) Is it decidable whether a quadratic equation in F admits a solution?

27. (source) Let $C_1$ and $C_2$ be distinct language classes such that $C_1$ is a subclass of $C_2$. Does there exist a reasonable group theoretic property P such that a group $G ∈epiC_1$ is equivalent to G has P and $G ∈epiC_2$?

28. (source) Does the semigroup $\text{End}(\mathbb{N}, <)$ admit a finest Polish semigroup topology?

29. (source) What more can be said about the class of groups which have all finitely generated simple vigorous groups as quotients?

30. (source) If two UDAF digraphs are shift equivalent then are they necessarily strong UDAF equivalent?

31. (source) Are there finitely presented non-virtually abelian groups with linear automorphic growth?

32. (source) If $G$ is a finitely generated simple group of homeomorphisms which satisfies Epstein’s Axioms, must $G$ actually be 2-generated?

33. (source) What is the automorphic growth of Thompson’s group F?

34. (source) Is the Diophantine problem decidable in Thompson’s group V?

35. (source) Is the standard restricted wreath product of $C_2$ with the Lamplighter group, an epiRegular group?

36. (source) Suppose that $M_1$ and $M_2$ are UDAF relator equivalent UDAF relator matrices with the same size. It is necessarily possible to convert $M_1$ into $M_2$ as in Theorem 7.8 without using operation (1)?

Answered Questions

(source) Are the two minimal Polish semigroup topologies and the unique maximal Polish semigroup topology on $I_{\mathbb{N}}$ the only three Polish semigroup topologies on $I_{\mathbb{N}}$?

Answer: No. This was first shown here. The paper classifies all Polish semigroup topologies on $I_{\mathbb{N}}$. There are countably infinitely many.

(source) Is the semigroup version of the Zariski topology applied to a group equal to the group version of the Zariski topology applied to the same group?

Answer: No. This was first shown here. Moreover here and here we see that Thompson’s group F is a counterexample.

(source) Is every countable Polish semigroup topologically isomorphic to a subsemigroup of $\mathbb{N}^{\mathbb{N}}$?

Answer: No. This was first shown here. Moreover the countable semigroup can even be chosen to be commutative, Clifford and locally compact.

(source) Is there an ω-categorical relational structure $A$ such that the topology of pointwise conver- gence on $\text{End}(A)$ is strictly finer than the Zariski topology?

Answer: Yes. This was first answered here. The structure can even be chosed to be homogeneous in a finite language.

(source) Are the Thompson monoids $M_{k,r}$ as defined by Birget finitely presented?

Answer: Yes. The proof is to appear in a paper with Reinis Cirpons, Alex Levine, James Mitchell and myself.

(source) What is the semigroup Zariski topology of the symmetric inverse monoid?

Answer: In a to appear paper with Serhii Bardyla, we show that this topology is zero-dimensional and every element which is not either $\mathscr{L}$ or $\mathscr{R}$ related to the identity has an uncountable neighbourhood base. We also gave a slightly better (but necessarily uncountable basis) for the topology. It’s ambiguous what it means to ask what a topology `is’ but I feel like this does that as much as is possible.

(source) Is the Frechet-Markov topology on a semigroup always contained in the Zariski topology?

Answer: No. In the same to appear paper with Serhii Bardyla we show that the semigroup Zarsiki topology on the symmetric inverse monoid is strictly courser than the Frechet-Markov topology.