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We compute explicitly the automorphism and outer automorphism group of all large-type free-of-infinity Artin groups. Our strategy involves reconstructing the associated Deligne complexes in a purely algebraic manner, i.e. in a way that is…
The classical subset construction for non-deterministic automata can be generalized to other side-effects captured by a monad. The key insight is that both the state space of the determinized automaton and its semantics---languages over an…
In this work we are going to study the dynamics of the linear automorphisms of a measure convolution algebra over a finite group, $T(\mu)=\nu * \mu$. In order to understand an classify the asymptotic behavior of this dynamical system we…
We study finite subgroups of outer automorphisms of free products. We give upper bounds for the orders of these finite subgroups as well as bounds for the orders of individual torsion outer automorphisms under some (necessary) conditions…
We describe the endomorphisms of the direct product of two free groups of finite rank and obtain conditions for which the subgroup of fixed points is finitely generated and we do the same for periodic points. We also describe the…
Two new classes of finite automata, called General hexagonal Boustrophedon finite automata and General hexagonal returning finite automata operating on hexagonal grids, are introduced and analyzed. The work establishes the theoretical…
Automatic groups admitting prefix closed automatic structures with uniqueness are characterized as the quotients of free groups by normal subgroups possessing sets of free generators satisfying certain language-theoretic conditions.
For discrete measured groupoids preserving a probability measure we introduce a notion of sofic dimension that measures the asymptotic growth of the number of sofic approximations on larger and larger finite sets. In the case of groups we…
We investigate the structure of the monoid of endomorphisms of the ordered set $(\mathbb{Q},{\leq})$ of rational numbers. We show that for any countable linearly ordered set $\Omega$, there are uncountably many maximal subgroups of…
This survey is intended to be a fast (and reasonably updated) reference for the theory of Stallings automata and its applications to the study of subgroups of the free group, with the main accent on algorithmic aspects. Consequently,…
We introduce a machine-learning approach (denoted Symmetry Seeker Neural Network) capable of automatically discovering discrete symmetry groups in physical systems. This method identifies the finite set of parameter transformations that…
The theory of finite automata concerns itself with words in a free monoid together with concatenation and without further structure. There are, however, important applications which use alphabets which are structured in some sense. We…
We study the class of groups generated by automata that act essentially freely on the boundary of a rooted tree. In the process we establish and discuss some general tools for determining if a group belongs to this class, and explore the…
Profinite semigroups are a generalization of finite semigroups that come about naturally when one is interested in considering free structures with respect to classes of finite semigroups. They also appear naturally through dualization of…
We explicitly determine the automorphism groups of all self-similar trees (a.k.a. trees with finitely many cone types). We show that any such automorphism group is a direct limit of certain finite products of finite symmetric groups, which…
A number of properties of spherical Artin groups extend to Garside groups, defined as the groups of fractions of monoids where least common multiples exist, there is no nontrivial unit, and some additional finiteness conditions are…
We study the Holomorphic and Random Dynamics of some rank 2 free groups generated by two H\'enon type maps. For these simply constructed examples we prove that the Fatou set is non-empty and that the stationary measures are supported on a…
Finite quasi semimetrics on $n$ can be thought of as nonnegative valuations on the edges of a complete directed graph on $n$ vertices satisfying all possible triangle inequalities. They comprise a polyhedral cone whose symmetry groups were…
In order to study how well a finite group might be generated by repeated random multiplications, P. Diaconis suggested the following urn model. An urn contains some balls labeled by elements which generate a group G. Two are drawn at random…
We introduce a new model of random Artin groups. The two variables we consider are the rank of the Artin groups and the set of permitted coefficients of their defining graphs. The heart of our model is to control the speed at which we make…