Conjugacy Class Growth in Virtually Abelian Groups

We show that every finitely generated group G with an element of order at least $(5rank(G))^{12}$ admits a locally finite directed Cayley graph with automorphism group equal to G. If moreover G is not generalized dihedral, then the above Cayley directed graph does not have bigons. On the other hand, if G is neither generalized dicyclic nor abelian and has an element of order at least $(2rank(G))^{36}$, then it admits an undirected Cayley graph with automorphism group equal to G. This extends classical results for finite groups and free products of groups. The above results are obtained as corollaries of a stronger form of rigidity which says that the rigidity of the graph can be observed in a ball of radius 1 around a vertex. This strong rigidity result also implies that the Cayley (di)graph covers very few (di)graphs. In particular, we obtain Cayley graphs of Tarski monsters which essentially do not cover other quasi-transitive graphs. We also show that a finitely generated group admits a locally finite labelled unoriented Cayley graph with automorphism group equal to itself if and only if it is neither generalized dicyclic nor abelian with an element of order greater than 2.

In any Coxeter group, the conjugates of elements in the standard minimal generating set are called reflections, and the minimal number of reflections needed to factor a particular element is called its reflection length. In this article we prove that the reflection length function on an affine Coxeter group has a uniform upper bound. More precisely, we prove that the reflection length function on an affine Coxeter group that naturally acts faithfully and cocompactly on ? n is bounded above by 2n, and we also show that this bound is optimal. Conjecturally, spherical and affine Coxeter groups are the only Coxeter groups with a uniform bound on reflection length.

Characterization of cyclically fully commutative elements in finite and affine Coxeter groups

Cannon I-3] has defined the property of almost convexity for the Cayley graph of a group with a given generating set. One takes the combinatorial ball of radius n in the Cayley graph and asks the following question: If two vertices in this ball are distance 2 from each other in the Cayley graph, how far must we travel inside the ball of radius n to get from one to the other? If this number is bounded independent of n, the group is said to be almost convex. Almost convexity relates metric properties of the Cayley graph to the efficiency with which that graph can be constructed, and therefore to the efficiency of algorithms for calculating in the group. The following sorts of groups are known I-3] to be almost convex: finite groups, HNN extensions and free products with amalgamation of finite groups, free groups, abelian groups, co-compact hyperbolic groups, and small cancellation groups. Recently Cannon, Floyd, Grayson and Thurston 1-5] have shown that groups acting as Solv geometry isometrics are not almost convex. Of Thurston's eight geometries [7], this leaves Nil and PSL2R. We will show that in certain generating sets, discrete co-compact groups of Nil isometries are almost convex. We will also see how to compute growth functions in these generating sets. Benson 1-1] has shown how to compute the growth for N z with respect to the generating set {x, y} (see below) x. Our method is more explicit, and we are able to extend the result to additional groups. The key to our approach is that

We show that for finite-range, symmetric random walks on general transient Cayley graphs, the expected occupation time of any given ball of radius r is O(r 5/2 ). We also study the volume-growth property of the wired spanning forests on general Cayley graphs, showing that the expected number of vertices in the component of the identity inside any given ball of radius r is O(r 11/2 ).

We investigate the so-called dual Matsumoto property or Hurwitz action in finite, affine and arbitrary Coxeter groups. In particular, we want to investigate how to reduce reflection factorizations and how two reflection factorizations of the same element are related to each other. We are motivated by the dual approach to Coxeter groups proposed by Bessis (2003) and the question whether there is an analogue of the well-known Matsumoto property for reflection factorizations. Our aim is a substantial understanding of the Hurwitz action. We therefore reprove uniformly results of Lewis–Reiner as well as Baumeister–Gobet–Roberts and the first author on the Hurwitz action in finite and affine Coxeter groups. Further we show that in an arbitrary Coxeter group all reduced reflection factorizations of the same element appear in the same Hurwitz orbit after a suitable extension by simple reflections. As parabolic quasi-Coxeter elements play an outstanding role in the study of the Hurwitz action, we aim to characterize these elements. We provide a characterization of maximal parabolic quasi-Coxeter elements in arbitrary Coxeter groups as well as a characterization of all parabolic quasi-Coxeter elements in affine Coxeter groups.

Associated with the Cayley graph (Γ, G) of a cancellative monoid Γ with a finite generating system G , we introduce two compact spaces: Ω(Γ, G) consisting of pre-partition functions and Ω(P_{Γ,G}) consisting of series opposite to the growth function P_{Γ,G}(t) := \sum^∞ _{n=0}\#Γ_n · t^n (where Γ_n is the ball of radius n centered at the unit element in the Cayley graph). Under mild assumptions on (Γ, G) , we introduce a fibration π : Ω(Γ, G) → Ω(P_{Γ,G}) , which is equivariant with respect to a (\tilde τ, τ) -action. The action is transitive if it is of finite order. Then we express the finite sum of the pre-partition functions in each fiber of π as a linear combination of the ratios of the residues of the two growth functions P_{Γ,G}(t) and P_{Γ,G}\mathcal M(t) := \sum^∞ {n=0}\mathcal M(Γ_n)t^n (where \mathcal M(Γ_n)/\#Γ_n is the free energy of the ball Γ_n ) at the poles on the circle of their convergence radius.

仿射Coxeter群（-B3,S）可以被看做仿射Coxeter群（-D4,-S）在满足条件α（-S）=-S的某种群自同构α下的不动点集合,设-l是-D4的长度函数,本文明显地刻画了加权Coxeter群（-B3,-l）的所有左胞腔.同时证明了：加权Coxeter群（-D4,-l）和（-B3,-l）的所有左胞腔都是左连通的,所有双边胞腔都是双边连通的。

In this paper, we are interested in the asymptotic enumeration of bipartite Cayley digraphs and Cayley graphs over abelian groups. Let be an abelian group and let be the automorphism of defined by , for every . A Cayley graph is said to have an automorphism group as small as possible if . In this paper, we show that, except for two infinite families, almost all bipartite Cayley graphs on abelian groups have automorphism group as small as possible. We also investigate the analogous question for bipartite Cayley digraphs. These results are used for the asymptotic enumeration of bipartite Cayley digraphs and graphs over abelian groups.

The main goal of the paper is to show that the fully commutative elements in the affine Coxeter group C ~ n \widetilde {C}_{n} form a union of two-sided cells. Then we completely answer the question of when the fully commutative elements of W W form or do not form a union of two-sided cells in the case where W W is either a finite or an affine Coxeter group.

The transposition network Tn of order n! is the Cayley graph of the symmetric group Sn with generators the set of all transpositions in Sn. Finding the bisection width of the transposition network is an open question posed by F. T. Leighton. We resolve this question for n even, by showing that the bisection width of the transposition network Tn is equal to nn!/4. When n ≥ 2 is odd, we show that the bisection width of the transposition network Tn is (1 + o(1))nn!/4. In doing so, we determine the second smallest eigenvalue of the adjacency matrix of Tn. Further, given a Cayley graph of a finite group G, with m conjugacy classes and with a set of generators closed under taking conjugates and not containing the identity of G, we show how to construct an m × m integer matrix Q, from the conjugacy classes of G, such that the set of eigenvalues of Q is equal to the set of eigenvalues of the adjacency matrix A of the given Cayley graph. Hence, when the order of G is large compared to the number of its conjugacy classes, a faster method for computing the eigenvalues of A is provided. © 1997 John Wiley & Sons, Inc.

An element of a Coxeter group $$W$$ is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were extensively studied by Stembridge, in particular in the finite case. They index naturally a basis of the generalized Temperley–Lieb algebra. In this work we deal with any finite or affine Coxeter group $$W$$ , and we give explicit descriptions of fully commutative elements. Using our characterizations we then enumerate these elements according to their Coxeter length, and find in particular that the corrresponding growth sequence is ultimately periodic in each type. When the sequence is infinite, this implies that the associated Temperley–Lieb algebra has linear growth.

The automorphism groups of Cayley graphs on symmetric groups, Cay(G, S), where S is a complete set of transpositions have been determined. In a similar spirit, automorphism groups of Cayley graphs Cay(An , S) on alternating groups An , where S is a set of all 3-cycles have also been determined. It has, in addition, been shown that these graphs are not normal. In all these Cayley graphs, one observes that their corresponding Cayley sets are a union of conjugacy classes. In this paper, we determine in their generality, the automorphism groups of Cay(G, S), where G ∈ {An , Sn } and S is a conjugacy class type Cayley set. Further, we show that the family of these graphs form a Boolean algebra. It is first shown that Aut(Cay(G, S)), S ∉ {∅, G \\ {e}}, is primitive if and only if G = An . Using one of the results obtained by Praeger in 1990, we exploit further the other cases, thereby proving that, for n > 4 and n ≠ 6, Aut(Cay(An , S)) ≅ Hol(An ) ⋊ 2, with Hol(G) ∼= G ⋊ Aut(G), provided that S is preserved by the outer automorphism defined by the conjugation by an odd permutation. Finally, in the remaining case G = Sn , n > 4 and n ≠ 6, we show that Aut(Cay(Sn, S) ≅ (Hol(An ) ⋊ 2) ≀ S 2 for S ⊂ An \\ {e}, and that Aut(Cay(Sn , S)) ≅ Hol(Sn ) ⋊ 2 otherwise; provided that S does not contain Sn \\ An or S ≠ An \\ {e}, S ∉ {∅, Sn \\ {e}}.

In this paper we study growth functions of automatic and hyperbolic groups. In addition to standard growth functions, we also want to count the number of finite graphs isomorphic to a given finite graph in the ball of radius n around the identity element in the Cayley graph. This topic was introduced to us by K. Saito [1991]. We report on fast methods to compute the growth function once we know the automatic structure. We prove that for a geodesic automatic structure, the growth function for any fixed finite connected graph is a rational function. For a word-hyperbolic group, we show that one can choose the denominator of the rational function independently of the finite graph.

A remark on the connectedness of spheres in Cayley graphs