arXiv subjects

Luna Elliott

Publications and source records attributed to Luna Elliott.

11 recordsLinked to original sources

arXiv2026

Polish topologies on endomorphism monoids of linear orders

In this paper, we investigate Polish semigroup topologies on the endomorphism monoids $\operatorname{End}(\mathbb{N},\leq)$ and $\operatorname{End}(\mathbb{Z},\leq)$. We introduce a new structural condition, property $\mathbb{XX}$, which yields automatic continuity of Borel measurable homomorphisms between certain topological semigroups. This provides a new method for analyzing Polish semigroup topologies on monoids with small groups of units. We show that for all monoids considered, the semigroup Zariski topology coincides with the pointwise topology and is therefore the coarsest Hausdorff semigroup topology. We prove that the submonoid $\operatorname{End}^{\infty}(\mathbb{N},\leq)$ of $\operatorname{End}(\mathbb{N},\leq)$ consisting of all endomorphisms with infinite image admits a unique Polish semigroup topology, namely the pointwise topology. On the other hand, despite possessing a finest Polish semigroup topology, the monoids $\operatorname{End}(\mathbb{N},\leq)$ and $\operatorname{End}(\mathbb{Z},\leq)$, admit infinitely many distinct Polish semigroup topologies. Also, we show that the monoid $\operatorname{End}(\mathbb{N},<)$ admits exactly $2^{\aleph_0}$ Polish semigroup topologies and no maximal second-countable semigroup topology.

math.GR↗

arXiv2026

Commutative decomposition of infinite symmetric groups and transformation monoids

The commutative subgroup width of a group $G$ is the smallest $k$ such that there are abelian subgroups $A_0,A_1,\ldots,A_{k-1}\leq G$ with $G=A_0A_1\cdots A_{k-1}$. Commutative (inverse) submonoid width is defined analogously. In 2002, Ab\'{e}rt showed, rather surprisingly, that the commutative subgroup width of the symmetric group on an infinite set is always finite. It was later shown by Seress that it is always bounded above by $14$. We answer a question of Seress and show that in fact the commutative subgroup width of $\operatorname{Sym}(\mathbb{N})$ is at most $9$. We improve the best known lower bound to $4$. We also study standard monoid analogues of the symmetric group; showing that the commutative submonoid widths of the full transformation monoid $\mathbb{N}^\mathbb{N}$, the partial transformation monoid $P_\mathbb{N}$ and the symmetric inverse monoid $I_\mathbb{N}$ are exactly $3$. We conclude by showing that the commutative inverse submonoid width of any infinite symmetric inverse monoid is always infinite.

math.GR↗

arXiv2026

Counting automorphic orbits in finitely generated groups

We study an analogue of the conjugacy growth function in finitely generated groups: the automorphic growth function. This counts the number of automorphic orbits that intersect the ball of radius $n$ in the group. We show that this is not a commensurability invariant, by giving virtually abelian counterexamples. We classify the automorphic growth rate of all virtually abelian groups of rank at most $2$, the Heisenberg group, finite rank free groups and Thompson's groups $T$ and $V$. This last computation allows to conclude that $T$ and $V$ have exponential conjugacy growth.

math.GR↗

arXiv2026

The Zariski Topology on Homeomorphism groups

The Zariski topology on a group G is the coarsest topology such that all sets of the form $\{x \in G | 1_G \neq g_0 x^{k_0} g_1 ... g_{l-1} x^{k_{l-1}} g_l\}$ are open. Originally introduced by Bryant as the verbal topology, it serves as a fundamental tool for investigating the topological structure of infinite groups and is always a $T_1$ topology with continuous shifts and inversion. Since the Zariski topology is coarser than every Hausdorff group topology on G, it provides a natural starting point for topologizing groups; specifically, for countable or abelian groups, it is known that the Zariski topology coincides with the Markov topology-the intersection of all Hausdorff group topologies on G. In this paper, we analyze the Zariski topology on various homeomorphism groups. We demonstrate that for the Thompson groups F and T, the Zariski (and thus Markov) topology coincides with the standard compact-open topology derived from their respective actions on $[0,1]$ and $S^1$. In contrast, we show that the Zariski (and thus Markov) topology on Thompson's group V is irreducible, and therefore neither Hausdorff nor a group topology. As V acts highly transitively on each of its orbits, this result stands in notable opposition to a theorem by Banakh et al, which establishes that the Zariski topology on any permutation group containing all finitely supported elements is a Hausdorff group topology. Our results for the Zariski topologies on $F,T$ and $V$ also apply to the full homeomorphism groups $\operatorname{Homeo}([0,1])$, $\operatorname{Homeo}(S^1)$, and $\operatorname{Homeo}(2^\omega)$ respectively. We conclude by providing a classification of the connected manifolds $M$ for which the homeomorphism group $\mathrm{Homeo}(M)$ admits a Hausdorff Zariski topology.

math.GR↗

arXiv2026

On constructing topology from algebra

In this thesis we explore natural procedures through which topological structure can be constructed from specific semigroups. We will do this in two ways: 1) we equip the semigroup object itself with a topological structure, and 2) we find a topological space for the semigroup to act on continuously. We discuss various minimum/maximum topologies which one can define on an arbitrary semigroup (given some topological restrictions). We give explicit descriptions of each these topologies for the monoids of binary relations, partial transformations, transformations, and partial bijections on a countable set. Using similar methods we determine whether or not each of these semigroups admits a unique Polish semigroup topology. We also do this for the various other semigroups, provide a proof of Rubin's theorem, and give a description of the automorphism groups of the Brin-Thompson groups. The thesis also contains many background results.

math.GR↗

arXiv2025

The Diophantine problem in Thompson's group F

We show that the Diophantine problem in Thompson's group F is undecidable. Our proof uses the facts that F has finite commutator width and rank 2 abelianisation, then uses similar arguments used by B\"uchi and Senger and Ciobanu and Garreta to show the Diophantine problem in free groups and monoids with abelianisation constraints is undecidable.

math.GR↗

arXiv2025

On epiC groups over language class C

We introduce a new framework linking group theory and formal language theory which generalizes a number of ways these topics have been linked in the past. For a language class C in the Chomsky hierarchy, we say a group is epiC if it admits a language $L$ over a finite (monoidal) generating set $X \subseteq G$ in the class C such that the image of L under the evaluation map is $G \setminus \{1_G\}$. We provide some examples of epiC groups and prove that the property of being epiC is not dependent on the generating set chosen. We also prove that epiC groups are closed under passage to finite index overgroups, taking extensions, and taking graph products of finitely many groups. Furthermore, we prove that epiRegular groups are closed under passage to finite index subgroups. Finally, we provide a characterization of the property of having solvable word problem within the framework of epiC groups.

math.GR↗

arXiv2024

Computing congruences of finite inverse semigroups

In this paper we present a novel algorithm for computing a congruence on an inverse semigroup from a collection of generating pairs. This algorithm uses a myriad of techniques from the theories of groups, automata, and inverse semigroups. An initial implementation of this algorithm outperforms existing implementations by several orders of magnitude.

math.GR↗

arXiv2024

E-disjunctive inverse semigroups

In this paper we provide an overview of the class of inverse semigroups $S$ such that every congruence on $S$ relates at least one idempotent to a non-idempotent; such inverse semigroups are called $E$-disjunctive. This overview includes the study of the inverse semigroup theoretic structure of $E$-disjunctive semigroups; a large number of natural examples; some asymptotic results establishing the rarity of such inverse semigroups; and a general structure theorem for all inverse semigroups where the building blocks are $E$-disjunctive.

math.GR↗

arXiv2024

Classifying the Polish semigroup topologies on the symmetric inverse monoid

We classify all Polish semigroup topologies on the symmetric inverse monoid on the natural numbers. This result answers a question of Elliott et al. There are countably infinitely many such topologies. Under containment, these Polish semigroup topologies form a join-semilattice with infinite descending chains, no infinite ascending chains, and arbitrarily large finite anti-chains. Also, we show that the monoid endowed with any second countable T_1 semigroup topology is homeomorphic to the Baire space.

math.RA↗

arXiv2023

Finite Presentability of Brin-Higman-Thompson Monoids via Free J\'onsson-Tarski Algebras

We show that the monoids totM_{k,1} introduced by Birget and their generalizations tot nM_{k,r} which extend the Brin-Higman-Thompson groups, can be realized as the endomorphism monoids of higher-dimensional J\'onsson-Tarski algebras. We also show how elements of these monoids can be thought of as "rewrite rules". We use these representations to show that the monoids are finitely presented.

math.RA↗