An Afternoon between Groups and Semigroups

Abstract: The study of conjugacy growth in finitely generated algebraic structures has a rich history, with applications ranging from the geometry of Riemannian manifolds to the combinatorics of formal languages. While the theory is

well-developed for groups, where conjugacy is uniquely defined, the monoid setting presents a new challenge: there is no single canonical notion of conjugacy.

In this talk we study conjugacy growth in the polycyclic monoids, a family of inverse monoids with zero that arise naturally in the theory of self-similar groups and directed graphs. We consider three natural conjugacy relations, which are strictly ordered by inclusion in polycyclic monoids and coincide on groups.

This is joint work with João Araújo, Wolfram Bentz, Michael Kinyon, Janusz Konieczny, and Valentin Mercier. 

Abstract: The plactic monoid is a well-known object, with connections to symmetric functions, representation theory, crystal bases, or Schubert calculus. In this talk, we consider two related ‘plactic-like’ structures: the hypoplactic monoid and the shifted plactic monoid. Estupiñán-Salamanca and Pechenik (2025) recently introduced an intrinsic characterization for the shifted plactic monoid via a universal property, and related it to a characterization of the plactic monoid due to Lascoux and Schützenberger (1981). Building on these works, we provide a characterization for the hypoplactic monoid, rephrasing a previous one by Novelli (2000), and establish relations between the three constructions. Joint work with António Malheiro. 

Abstract: For every language L over a finite alphabet A, there exists a deterministic, complete, and accessible automaton that is, in a certain way, minimal among all the automata recognizing that language. The set of states of the minimal automaton can be characterized as the set of all characteristic functions of the left quotients of the language with respect of sets containing only one word. In this way, we may interpret the minimal automaton of any language as a subset of   {0,1}^A*.

In this context, by considering {0,1} as a discrete space and giving {0,1}^A* the product topology, the minimal topological automaton of a language can be seen as the closure of the corresponding minimal automaton in the topological space  {0,1}^A*.

When we restrict to the case where the alphabet A contains only a single letter, the minimal topological automaton can be interpreted as a one-sided shift space, that is, a compact subspace of the Cantor space that is closed under the shift map. Shift spaces are a widely studied object, and in this seminar we will explore their relation with topological automata, and see some results that arise from this connection. Moreover, we will relate these concepts to certain types of binary trees. We define a binary tree as a subset of {0,1}* that is closed under prefixes.

Abstract: In this talk we investigate the conditions under which a given circular (synchronizing) DFA is simple (sometimes referred to as primitive) and when it is irreducible. Our notion of irreducibility slightly differs from the classical one, since we are considering our monoid representations to be complex; nevertheless, several well-known results remain valid—for instance, the fact that every irreducible automaton is necessarily simple. We provide a complete characterization of simplicity in the circular case by means of the so-called weak contracting property. Furthermore, we establish necessary and sufficient conditions for a circular contracting automaton (a stronger condition than the weakly contracting one) to be irreducible, and we present examples illustrating our results. This work is motivated by the conjecture stating that extremal automata (in view of the Černý conjecture) are all irreducible, and we believe that the techniques adopted here can be generalized to the non-circular case.

Abstract:  In group theory, the automorphism tower problem is a well-known problem asking whether starting from a group G, the sequence of automorphism groups terminates in a finite number of steps. A classical result from Weilant states that this is the case for any finite group, while some other results show that this condition also hold for certain classes of infinite groups.

We can naturally look at the analogous problem in the realm of semigroup theory by considering endomorphism monoids in place of automorphism groups, which would lead to the following formulation: given a monoid M, does the sequence of endomorphism monoids End(M), End(End(M)), ... stabilises in a finite number of steps? This question on the endomorphism tower of M is easily resolved in the negative when M is a non-trivial finite monoid. In this talk, we will then look at another question linking the two earlier ones: given a monoid M, is there a "nice" pattern in the sequence of groups Aut(End(M)), Aut(End(End(M))), ...? In particular, we will focus on the case where M is a finite symmetric group Sn (with n large enough), and show that the first automorphism groups in the endomorphism tower are all isomorphic to Sn.

Abstract: In this talk we will discuss a language-theoretic approach to study the density of subsets in nonabelian free groups. Concretely, we will show how an infinite monkey theorem-like result allows us to prove that automorphic orbits of elements in free groups have density 0. We will then discuss the case of  finitely generated subgroups of positive density, where the situation can be fully understood.