Related papers: Conjugacy growth series and languages in groups

We show that real tight frames that generate lattices must be rational, and use this observation to describe a construction of lattices from vertex transitive graphs. In the case of irreducible group frames, we show that the corresponding…

There exist very lucid explanations of the combinatorial origins of rational and algebraic functions, in particular with respect to regular and context free languages. In the search to understand how to extend these natural correspondences,…

Geometric grid classes of permutations have proven to be key in investigations of classical permutation pattern classes. By considering the representation of gridded permutations as words in a trace monoid, we prove that every geometric…

There is a well-known classification of conjugacy classes of involutions in finite Coxeter groups, in terms of subsets of nodes of their Coxeter graphs. In many cases, the product of an involution with the longest element is again an…

K. Harada conjectured for any finite group $G$, the product of sizes of all conjugacy classes is divisible by the product of degrees of all irreducible characters. We study this conjecture when $G$ is the general linear group over a finite…

This paper is concerned with what intermediate syntactic categories children acquire during first language development, and in what order. Maturational theories make different predictions. Bottom-up accounts (GROWING) propose that lexical…

The geodesic total curvature of rectifiable spherical curves is analyzed. We extend to the case of high dimension spheres the explicit formula that holds true for curves supported into the 2-sphere. For this purpose, we take advantage of…

I describe a class of groups acting on rooted trees. The original claim was that all have intermediate word growth between polynomial and exponential. The argument constructs a functional equation on the growth formal power series, and…

Let $X$ be a CAT(0) cubical complex. The growth series of $X$ at $x$ is $G_{x}(t)=\sum_{y \in Vert(X)} t^{d(x,y)}$, where $d(x,y)$ denotes $\ell_{1}$-distance between $x$ and $y$. If $X$ is cocompact, then $G_{x}$ is a rational function of…

We study growth rates of generalised Fibonacci sequences of a particular structure. These sequences are constructed from choosing two real numbers for the first two terms and always having the next term be either the sum or the difference…

In this paper, we present a new technique for studying the dynamics of a finitely generated rational semigroup. Such a semigroup can be associated naturally to a certain holomorphic correspondence on $\mathbb{P}^1$. Then, results on the…

In this paper we exhibit two infinite families of trees $\{T^1_n\}_{n \geq 17}$ and $\{T^2_n\}_{n \geq 17}$ on $n$ vertices, such that $T^1_n$ and $T^2_n$ are non-isomorphic, co-spectral, and the right-angled Coxeter groups (RACGs) based on…

We develop a theory of soficity for actions on graphs and obtain new applications to the study of sofic groups. We establish various examples, stability and permanence properties of sofic actions on graphs, in particular soficity is…

This paper measures the impact of increased exposure on whether learned construction grammars converge onto shared representations when trained on data from different registers. Register influences the frequency of constructions, with some…

The conjugacy problem belongs to algorithmic group theory. It is the following question: given two words x, y over generators of a fixed group G, decide whether x and y are conjugated, i.e., whether there exists some z such that zxz^{-1} =…

The fundamental group of the complement of a hyperplane arrangement plays an important role in studying the corresponding arrangements. In particular, for large families of hyperplane arrangements, this fundamental group, being isomorphic…

A subset S of a group G invariably generates G if G = <s^(g(s)) | s in S> for each choice of g(s) in G, s in S. In this paper we study invariable generation of infinite groups, with emphasis on linear groups. Our main result shows that a…

Let $G$ be a toral relatively hyperbolic group, and let $\varphi\in\mathrm{Aut}(G)$. We prove that, under iteration of $\varphi$, the conjugacy length $||\varphi^n(g)||$ of every element $g\in G$ grows like $n^d\lambda^n$ for some…

Jordan Normal Forms serve as excellent representatives of conjugacy classes of matrices over closed fields. Once we knows normal forms, we can compute functions of matrices, their main invariant, etc. The situation is much more complicated…

The natural language generation (NLG) component of a spoken dialogue system (SDS) usually needs a substantial amount of handcrafting or a well-labeled dataset to be trained on. These limitations add significantly to development costs and…