arXiv research

Abstract Coxeter groups have growth rates that are Perron numbers.

Word metrics from large balls in injective spaces are exponentially generic and growth tight.

Estimates growth of reciprocal classes in Hecke groups.

Growth rates of geodesics on modular orbifolds are studied.

Let Mod(S) denote the mapping class group of a compact, orientable surface S. We prove that finitely generated subgroups of Mod(S) which are not virtually abelian have uniform exponential growth with minimal growth rate bounded below by a constant depending only, and necessarily, on S. For the proof, we find in any suc…

A simple text model shows word lengths follow Zipf's law.

Lehmer's question is equivalent to one about generalized growth rates of Lefschetz numbers of iterated pseudo-Anosov surface homeomorphisms. One need consider only homeomorphisms that arise as monodromies of fibered knots in lens spaces L(n,1), n>0. Lehmer's question for Perron polynomials is equivalent to one about ge…

Let $Σ$ be a surface of negative Euler characteristic and $S$ a generating set for $π_1(Σ,p)$ consisting of simple loops that are pairwise disjoint (except at $p$). We show that the word length with respect to $S$ of an element of $π_1(Σ,p)$ is given by its intersection number with a well-chosen collection of curves an…

Let $G$ be a finitely generated group with a finite generating set $S$. For $g\in G$, let $l_S(g)$ be the length of the shortest word over $S$ representing $g$. The growth series of $G$ with respect to $S$ is the series $A(t) = \sum_{n=0}^\infty a_n t^n$, where $a_n$ is the number of elements of $G$ with $l_S(g)=n$. If…

Extends growth properties of hyperbolic groups to their extensions.

The paper finds formulas for word lengths and conjugacy classes in surface groups.

The study examines the growth of reciprocal classes in Hecke groups, proving an asymptotic formula.

We give a quantification of residual finiteness for the fundamental groups of hyperbolic manifolds that admit a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. Specifically, we give explicit upper bounds on residual finiteness that are linear in terms of geodesic length. We t…

We prove that the exponential growth rate of the regular language of penetration sequences is smaller than the growth rate of the regular language of normal form words, if the acceptor of the regular language of normal form words is strongly connected. Moreover, we show that the latter property is satisfied for all irr…

The study examines complete Kähler manifolds with nonnegative Ricci curvature and discovers rigidity properties.

The study calculates the growth rate of reciprocal hyperbolic elements in Hecke groups.

We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, non-elementary group $G$ acting on a $G$--space $\mathcal{X}$, we prove that if $G$ contains a strongly contracting eleme…

The paper calculates the growth rates of billiard languages in hyperbolic polygons.

Study inert and ambiguous classes in modular group using combinatorial methods.

Study on quantitative aspects of trace polynomials in free groups.

We study a coarse homology theory with prescribed growth conditions. For a finitely generated group G with the word length metric this homology theory turns out to be related to amenability of G. We characterize vanishing of a certain fundamental class in our homology in terms of an isoperimetric inequality on G and sh…

For the free group $F_r$ on $r>1$ generators (respectively, the free product $G_1 * G_2$ of two nontrivial finite groups $G_1$ and $G_2$), we obtain the asymptotic for the number of conjugacy classes of commutators in $F_r$ (respectively, $G_1 * G_2$) with a given word length in a fixed set of free generators (respecti…

In the late 90's, after severe financial and economic crisis, accompanied by inflation and exchange rate instability, Eastern Europe emerged into two groups of countries with radically contrasting monetary regimes (Currency Boards and Inflation targeting). The task of our study is to compare econometrically the perform…

Following the financial crisis of the late 2000s, policy makers have shown considerable interest in monitoring financial stability. Several central banks now publish indices of financial stress, which are essentially based upon market related data. In this paper, we examine the potential for improving the indices by de…

Let $Γ$ be the fundamental group of a manifold modeled on three dimensional Sol geometry. We prove that $Γ$ has a finite index subgroup $G$ which has a rational growth series with respect to a natural generating set. We do this by enumerating $G$ by a regular language. However, in contrast to most earlier proofs of thi…

Let $Γ$ be a finitely generated discrete group satisfying the rapid decay condition. We give a new proof of the higher Atiyah-Patodi-Singer theorem on a Galois $Γ$-coverings, thus providing an explicit formula for the higher index associated to a group cocycle $c\in Z^k (Γ;\mathbb{C})$ which is of polynomial growth wit…

We show that an Anosov map has a geodesic axis on the curve graph of a torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any…

We establish the existence, finiteness, and uniqueness up to scaling of various isoperimetric profiles of a group, in all dimensions. We also show that these profiles all coincide in dimensions 4 and higher; in particular, the nth Dehn function is equal to FV^{n+1} for n at least 3. Even for dimension 3, there is signi…

Quasimorphisms and pseudo-Anosov flows

We give a new, effective proof of the separability of cubically convex-cocompact subgroups of special groups. As a consequence, we show that if $G$ is a virtually compact special hyperbolic group, and $Q\leq G$ is a $K$-quasiconvex subgroup, then any $g\in G-Q$ of word-length at most $n$ is separated from $Q$ by a subg…

Viewing Dehn's algorithm as a rewriting system, we generalise to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to include nilpotent groups, many relatively hyperbolic groups including geometrically…

Study growth patterns in random networks using i.i.d. perturbations.

Paper benchmarks Bengali language classification tasks using MConv-LSTM network.

Let $F_n$ be the free group on $n$ generators and $Γ_g$ the surface group of genus $g$. We consider two particular generating sets: the set of all primitive elements in $F_n$ and the set of all simple loops in $Γ_g$. We give a complete characterization of distorted and undistorted elements in the corresponding $Aut$-in…

Study of Bowditch representations in hyperbolic spaces with implications for dynamics and recognition.

Let $G$ be a finite group with symmetric generating set $S$, and let $c = \max_{R > 0} |B(2R)|/|B(R)|$ be the doubling constant of the corresponding Cayley graph, where $B(R)$ denotes an $R$-ball in the word-metric with respect to $S$. We show that the multiplicity of the $k$th eigenvalue of the Laplacian on the Cayley…

What is the role of social interactions in the creation of price bubbles? Answering this question requires obtaining collective behavioural traces generated by the activity of a large number of actors. Digital currencies offer a unique possibility to measure socio-economic signals from such digital traces. Here, we foc…

It is well known that the compactifications of the canonical contact systems living on real jet spaces $J^{\,r}(1,1)$, $r \ge 2$, are locally universal Goursat distributions, $Δ^r$, living on compact manifolds (called Goursat monsters) having open dense jet-like ($J^{\,r}(1,1)$-like) parts. By virtue of the results of …

Separable Non-negative Matrix Factorization (SNMF) is an important method for topic modeling, where "separable" assumes every topic contains at least one anchor word, defined as a word that has non-zero probability only on that topic. SNMF focuses on the word co-occurrence patterns to reveal topics by two steps: anchor…

A model is presented of the market dynamics to emphasis the effects of increasing returns to scale, including the description of the born and death of the adaptive producers. The evolution of market structure and its behavior with the technological shocks are discussed. Its dynamics is in good agreement with some empir…

Most existing word embedding approaches do not distinguish the same words in different contexts, therefore ignoring their contextual meanings. As a result, the learned embeddings of these words are usually a mixture of multiple meanings. In this paper, we acknowledge multiple identities of the same word in different co…

There are certain families of words and word sequences (words in the generators of a two-generator group) that arise frequently in the Teichm{ü}ller theory of hyperbolic three-manifolds and Kleinian and Fuchsian groups and in the discreteness problem for two generator matrix groups. We survey some of the families of su…

Geometrically transforms word embeddings into a common space for better comparison.

We discuss a topological approach to words introduced by the author. Words on an arbitrary alphabet are approximated by Gauss words and then studied up to natural modifications inspired by the Reidemeister moves on knot diagrams. This leads us to a notion of homotopy for words. We introduce several homotopy invariants …

Continuous word representation (aka word embedding) is a basic building block in many neural network-based models used in natural language processing tasks. Although it is widely accepted that words with similar semantics should be close to each other in the embedding space, we find that word embeddings learned in seve…

The abstract explains how word and relation representations capture semantic meaning.

Paper proposes MMD-Sense-Analysis for detecting word sense shifts.

Approaches KL divergence for learning multi-sense word distributions.