arXiv subjects

Motiejus Valiunas

Publications and source records attributed to Motiejus Valiunas.

21 records · Page 2Linked to original sources

arXiv2019

Rational growth and degree of commutativity of graph products

Let $G$ be an infinite group and let $X$ be a finite generating set for $G$ such that the growth series of $G$ with respect to $X$ is a rational function; in this case $G$ is said to have rational growth with respect to $X$. In this paper a result on sizes of spheres (or balls) in the Cayley graph $Γ(G,X)$ is obtained: namely, the size of the sphere of radius $n$ is bounded above and below by positive constant multiples of $n^αλ^n$ for some integer $α\geq 0$ and some $λ\geq 1$. As an application of this result, a calculation of degree of commutativity (d. c.) is provided: for a finite group $F$, its d. c. is defined as the probability that two randomly chosen elements in $F$ commute, and Antolín, Martino and Ventura have recently generalised this concept to all finitely generated groups. It has been conjectured that the d. c. of a group $G$ of exponential growth is zero. This paper verifies the conjecture (for certain generating sets) when $G$ is a right-angled Artin group or, more generally, a graph product of groups of rational growth in which centralisers of non-trivial elements are "uniformly small".

math.GR↗

arXiv2017

Negligibility of parabolic elements in relatively hyperbolic groups

We study density of parabolic elements in a finitely generated relatively hyperbolic group $G$ with respect to a word metric. We prove this density to be zero (apart from degenerate cases) and the limit defining the density to converge exponentially fast; this has recently been proven independently by W. Yang. As a corollary, we obtain the analogous result for the set of commuting pairs of elements in $G^2$, showing that the degree of commutativity of $G$ is equal to zero.

math.GR↗

arXiv2015

Orbits of crystallographic embedding of non-crystallographic groups and applications to virology

The architecture of infinite structures with non-crystallographic symmetries can be modeled via aperiodic tilings, but a systematic construction method for finite structures with non-crystallographic symmetry at different radial levels is still lacking. We present here a group theoretical method for the construction of finite nested point set with non-crystallographic symmetry. Akin to the construction of quasicrystals, we embed a non-crystallographic group $G$ into the point group $\mathcal{P}$ of a higher dimensional lattice and construct the chains of all $G$-containing subgroups. We determine the orbits of lattice points under such subgroups, and show that their projection into a lower dimensional $G$-invariant subspace consists of nested point sets with $G$-symmetry at each radial level. The number of different radial levels is bounded by the index of $G$ in the subgroup of $\mathcal{P}$. In the case of icosahedral symmetry, we determine all subgroup chains explicitly and illustrate that these point sets in projection provide blueprints that approximate the organisation of simple viral capsids, encoding information on the structural organisation of capsid proteins and the genomic material collectively, based on two case studies. Contrary to the affine extensions previously introduced, these orbits endow virus architecture with an underlying finite group structure, which lends itself better for the modelling of its dynamic properties than its infinite dimensional counterpart.

math-ph↗