Saturday, June 4th

• 10:00 - 11:00 Oleg Bogopolski (Szczecin University)

Structure of solutions of exponential equations in acylindrically hyperbolic groups

An exponential equation over a group G is an equation of the form a_1g_1^{x_1}a_2g_2^{x_2}... a_ng_n^{x_n}=1, where

a_1,g_1,..., a_n,g_n are elements from G (called coefficients) and x_1,\dots,x_n are variables which take values in Z.

In the first part of my talk I'll survey some known results about solutions of exponential equations in different classes of groups and review open problems. In the second part I'll explain results from my recent preprint with A. Bier:

Let G be a group acting acylindrically on a hyperbolic space and let E be an exponential equation over G. We show that E is equivalent to a finite disjunction of finite systems of pairwise independent equations which are either loxodromic over virtually cyclic subgroups or elliptic. We also obtain a description of the solution set of E. We obtain stronger results in the case where G is hyperbolic relative to a collection of peripheral subgroups \{H_i\}_{i\in I}. In particular, we prove in this case that the solution sets of exponential equations over G are Z-semilinear if and only if the solution sets of exponential equations over every H_i are Z-semilinear.
We obtain an analogous result for finite disjunctions of finite systems of exponential equations and inequations over relatively hyperbolic groups in terms of definabe sets in the weak Presburger arithmetic.

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• 11:15 - 12:15 Alan Logan (University of St Andrews)

The conjugacy problem for ascending HNN-extensions of free groups

In this talk, I will explain how to solve the conjugacy problem for ascending HNN-extensions of free groups. In 2006, Bogopolski+Martino+Maslakova+Ventura solved the conjugacy problem for free-by-cyclic groups. Their proof is based on 2 key components, which are both proven using an analysis of free groups automorphisms via train-track maps. We follow this same route, but instead use an analysis of free group endomorphisms via the "automorphic expansions" of Mutanguha to prove the analogous 2 key components.

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• 12:30 - 13:30 Emanuele Rodaro (Politecnico di Milano)

Tree automaton groups, reducible automata, and poly-context-free groups

(Semi)groups defined by the action of Mealy transducers have become very popular after the introduction of the Grigorchuk group as the first example of a group with intermediate growth. This class of groups and semigroups have deep connections with many areas of mathematics, from the theory of profinite groups to complex dynamics and theoretical computer science, and they serve as a source of examples and counterexamples for many important group theoretic problems. In this talk, I will consider a construction, recently introduced in collaboration with M.Cavaleri, A.Donno, and D. D'Angeli, that associates a certain automaton (semi)group to a finite simple graph. I will then focus on automaton groups associated with trees and a possible generalisation, called reducible automaton groups, for which we give a general structure theorem that shows that all reducible automaton groups are direct limit of poly-context-free groups which are virtually subgroups of the direct product of free groups. This result partially supports a conjecture by T. Brough regarding the structure of poly-context-free groups, a natural generalisation of virtually free groups.