Geodesic Growth Research Articles

Self-Avoiding Walks on Cayley Graphs Through the Lens of Symbolic Dynamics

We study dynamical and computational properties of the set of bi-infinite self-avoiding walks on Cayley graphs, as well as ways to compute, approximate and bound their connective constant. To do this, we introduce the skeleton $X_{G,S}$ of a finitely generated group $G$ relative to a generating set $S$, which is a one-dimensional subshift made of configurations on $S$ that avoid all words that reduce to the identity. We provide a characterization of groups which have SFT skeletons and sofic skeletons: first, there exists a finite generating set $S$ such that $X_{G,S}$ is a subshift of finite type if and only if $G$ is a plain group; second, there exists $S$ such that $X_{G,S}$ is sofic if and only if $G$ is a plain group, $\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ or $\mathcal{D}_{\infty}\times\mathbb{Z}/2\mathbb{Z}$. We also characterize finitely generated torsion groups as groups whose skeletons are aperiodic. For connective constants, using graph height functions and bridges, we show that Cayley graphs of finitely generated torsion groups do not admit graph height functions, and that for groups that admit transitive graph height functions, the connective constant is equal to the growth rate of periodic points of the skeleton. Finally, we take a brief look at the set of bi-infinite geodesics and introduce an analog of the connective constant for the geodesic growth.

Geodesic Growth of Numbered Graph Products

In this paper, we study geodesic growth of numbered graph products; these are a generalization of right-angled Coxeter groups, defined as graph products of finite cyclic groups. We first define a graph-theoretic condition called link-regularity, as well as a natural equivalence amongst link-regular numbered graphs, and show that numbered graph products associated to link-regular numbered graphs must have the same geodesic growth series. Next, we derive a formula for the geodesic growth of right-angled Coxeter groups associated to link-regular graphs. Finally, we find a system of equations that can be used to solve for the geodesic growth of numbered graph products corresponding to link-regular numbered graphs that contain no triangles and have constant vertex numbering.

Geodesic growth of some 3-dimensional RACGs

Geodesic growth of some 3-dimensional RACGs

A virtually 2-step nilpotent group with polynomial geodesic growth

A direct consequence of Gromov's theorem is that if a group has polynomial geodesic growth with respect to some finite generating set then it is virtually nilpotent. However, until now the only examples known were virtually abelian. In this note we furnish an example of a virtually 2-step nilpotent group having polynomial geodesic growth with respect to a certain finite generating set.

Growth Rates of Coxeter Groups and Perron Numbers

AbstractWe define a large class of abstract Coxeter groups that we call $\infty $–spanned, and for which the word growth rate and the geodesic growth rate appear to be Perron numbers. This class contains a fair amount of Coxeter groups acting on hyperbolic spaces, thus corroborating a conjecture by Kellerhals and Perren. We also show that for this class the geodesic growth rate strictly dominates the word growth rate.

Experiments on growth series of braid groups

We introduce an algorithmic framework to investigate spherical and geodesic growth series of braid groups relatively to the Artin's or Birman–Ko–Lee's generators. We present our experimentations in the case of three and four strands and conjecture rational expressions for the spherical growth series with respect to the Birman–Ko–Lee's generators.

Geodesic growth in virtually abelian groups

We show that the geodesic growth function of any finitely generated virtually abelian group is either polynomial or exponential; and that the geodesic growth series is holonomic, and rational in the polynomial growth case. In addition, we show that the language of geodesics is blind multicounter.

Spherical and geodesic growth rates of right-angled Coxeter and Artin groups are Perron numbers

We prove that for any infinite right-angled Coxeter or Artin group, its spherical and geodesic growth rates (with respect to the standard generating set) either take values in the set of Perron numbers, or equal 1. Also, we compute the average number of geodesics representing an element of given word-length in such groups.

Laplacian networks: Growth, local symmetry, and shape optimization.

Inspired by river networks and other structures formed by Laplacian growth, we use the Loewner equation to investigate the growth of a network of thin fingers in a diffusion field. We first review previous contributions to illustrate how this formalism reduces the network's expansion to three rules, which respectively govern the velocity, the direction, and the nucleation of its growing branches. This framework allows us to establish the mathematical equivalence between three formulations of the direction rule, namely geodesic growth, growth that maintains local symmetry, and growth that maximizes flux into tips for a given amount of growth. Surprisingly, we find that this growth rule may result in a network different from the static configuration that optimizes flux into tips.

Geodesic growth of right-angled Coxeter groups based on trees

In this paper, we exhibit two infinite families of trees $$\{T^1_n\}_{n \ge 17}$${Tn1}nź17 and $$\{T^2_n\}_{n \ge 17}$${Tn2}nź17 on n vertices, such that $$T^1_n$$Tn1 and $$T^2_n$$Tn2 are non-isomorphic, co-spectral, with co-spectral complements, and the right-angled Coxeter groups (RACGs) based on $$T^1_n$$Tn1 and $$T^2_n$$Tn2 have the same geodesic growth with respect to the standard generating set. We then show that the spectrum of a tree is not sufficient to determine the geodesic growth of the RACG based on that tree, by providing two infinite families of trees $$\{S^1_n\}_{n \ge 11}$${Sn1}nź11 and $$\{S^2_n\}_{n \ge 11}$${Sn2}nź11, on n vertices, such that $$S^1_n$$Sn1 and $$S^2_n$$Sn2 are non-isomorphic, co-spectral, with co-spectral complements, and the RACGs based on $$S^1_n$$Sn1 and $$S^2_n$$Sn2 have distinct geodesic growth. Asymptotically, as $$n\rightarrow \infty $$nźź, each set $$T^i_n$$Tni, or $$S^i_n$$Sni, $$i=1,2$$i=1,2, has the cardinality of the set of all trees on n vertices. Our proofs are constructive and use two families of trees previously studied by B. McKay and C. Godsil.

The growth rates for pure Artin groups of dihedral type

We consider the kernel of the natural projection from the Artin group of dihedral type I2(k) to the associated Coxeter group, which we call a pure Artin group of dihedral type and write PI2(k). We show that the growth rates for both the spherical growth series and geodesic growth series of PI2(k) with respect to a natural generating set are Pisot numbers.

Geodesic growth in right-angled and even Coxeter groups

The objective of this paper is to detect which combinatorial properties of a regular graph can completely determine the geodesic growth of the right-angled Coxeter or Artin group this graph defines, and to provide the first examples of right-angled and even Coxeter groups with the same geodesic growth series.

ON GROUPS WHOSE GEODESIC GROWTH IS POLYNOMIAL

This paper records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely generated group G has an element whose normal closure is abelian and of finite index, then G has a finite generating set with respect to which the geodesic growth is polynomial (this includes all virtually cyclic groups).

BUSEMANN POINTS OF ARTIN GROUPS OF DIHEDRAL TYPE

We study the horofunction boundary of an Artin group of dihedral type with its word metric coming from either the usual Artin generators or the dual generators. In both cases, we determine the horoboundary and say which points are Busemann points, that is, the limits of geodesic rays. In the case of the dual generators, it turns out that all boundary points are Busemann points, but this is not true for the Artin generators. We also characterize the geodesics with respect to the dual generators, which allows us to calculate the associated geodesic growth series.

GROWTH SERIES FOR ARTIN GROUPS OF DIHEDRAL TYPE

We consider the Artin groups of dihedral type I2(k) defined by the presentation Ak = 〈a,b | prod (a,b;k) = prod (b,a;k)〉 where prod (s,t;k) = ststs …, with k terms in the product on the right-hand side. We prove that the spherical growth series and the geodesic growth series of Ak with respect to the Artin generators {a,b,a-1, b-1} are rational. We provide explicit formulas for the series.

GRAPH PRODUCTS AND CANNON PAIRS

A pair (G, A) consisting of a group G and a finite generating set A is a Cannon pair if the language of all geodesics in the associated Cayley graph is regular. We prove that the Cannon pair property is preserved by graph products and indicate applications of this result to the geodesic and spherical growth series of graph products.

Complete growth functions of hyperbolic groups

. We study a generalization of the growth functions of finitely generated groups, namely the growth functions Σg∈Ggz|g| with coefficients in the group ring ℤ[G]. Rationality and methods of computation of such functions are discussed, in particular for hyperbolic groups. The complete growth functions of surface groups are explicitly computed. The operator and geodesic growth functions are also studied.