The normalized cyclomatic quotient associated with presentations of finitely generated groups

This chapter continues the development of group theory begun in Chapter IV, the main topics being the use of generators and relations, representation theory for finite groups, and group extensions. Representation theory uses linear algebra and inner-product spaces in an essential way, and a structure-theory theorem for finite groups is obtained as a consequence. Group extensions introduce the subject of cohomology of groups.Sections 1–3 concern generators and relations. The context for generators and relations is that of a free group on the set of generators, and the relations indicate passage to a quotient of this free group by a normal subgroup. Section 1 constructs free groups in terms of words built from an alphabet and shows that free groups are characterized by a certain universal mapping property. This universal mapping property implies that any group may be defined by generators and relations. Computations with free groups are aided by the fact that two reduced words yield the same element of a free group if and only if the reduced words are identical. Section 2 obtains the Nielsen-Schreier Theorem that subgroups of free groups are free. Section 3 enlarges the construction of free groups to the notion of the free product of an arbitrary set of groups. Free product is what coproduct is for the category of groups; free groups themselves may be regarded as free products of copies of the integers.Sections 4–5 introduce representation theory for finite groups and give an example of an important application whose statement lies outside representation theory. Section 4 contains various results giving an analysis of the space C(G, ?) of all complex-valued functions on a finite group G. In this analysis those functions that are constant on conjugacy classes are shown to be linear combinations of the characters of the irreducible representations. Section 5 proves Burnside’s Theorem as an application of this theory—that any finite group of order paqb with p and q prime and with a+b > 1 has a nontrivial normal subgroup.Section 6 introduces cohomology of groups in connection with group extensions. If N is to be a normal subgroup of G and Q is to be isomorphic to G/N, the first question is to parametrize the possibilities for G up to isomorphism. A second question is to parametrize the possibilities for G if G is to be a semidirect product of N and Q.

Expanders are highly connected sparse graphs of great interest in computer science, in areas ranging from parallel computation to complexity theory, from cryptography to coding theory, and, most recently, computational group theory (see e.g. [AKS, G+, LP, SS, V, WZ].) The explicit constructions of expander graphs [M1, M2, LPS] (see also [GG, Lu]) use deep mathematical tools to construct families of Cayley graphs of finite groups. The Independence Problem [LW], is whether being an expander family is a property of the groups alone, independent of the choice of generators. A counterexample to the general the problem was recently obtained in [ALW], by using a new combinatorial construction of expanders [RVW]. The problem remains for all classical series of finite simple groups (An, PSL(2,p), etc.)Let G be a finite group generated by at most d elements. The product replacement graph Γk(G) is defined to be a graph, with vertices corresponding to generating k-tuples in G, and edges corresponding to Nielsen transformations. Most recently, graphs Γk(G) became a subject of an intense investigation, prompted by the study of a commonly used 'practical' product replacement algorithm for generating random elements in finite groups, designed Leedham-Green and Soicher [LG]. This algorithm, based on the random walk on graphs Γk(G), showed a remarkable performance, as reported in [C+]. It was suggested in [LP], and proved in several special cases, that the product replacement graphs Γk(G) are expanders, for a fixed k, when |G| → ∞The main result of this paper is Theorem 2, establishing the connection between the expansion coefficient of the product replacement graph Γk(G) and the minimal expansion coefficient of a Cayley graph of group G with k generators. One can think of our result as of an additional evidence in favor of the speculation in [LP]. On the other hand, it gives an algorithmic motivation for study of the Independence Problem, in the aftermath of [ALW].In a special case (see Theorem 1 below), we show that if one assumes that all Cayley graphs with at most four generators in PSL (2, p) have a universal lower bound on expansion, then the product replacement graphs Γk(PSL(2,p)) form an expander family, when k ≥ 8 is fixed, and p → ∞.The proof of the Theorems consists of two parts: probability on groups and graph theoretic. The latter is based on a new combinatorial result of independent interest (Lemmas 4, 5), which generalizes a classical lemma that the direct product of expanders is an expander (see e.g. [Ch]). We prove that if a graph Γ on a direct product of sets satisfies two conditions:1) all 'row' subgraphs are expanders,2) at least (1-e) proportion of 'column' graphs are expanders,then graph &Gamma is also an expander. We use a novel combinatorial technique based on graph decomposition, as opposed to path arguments used in previously in [CG, DS2, DS3, P4]. It is easy to see that such a technique can never prove that a certain family of graphs is an expander family (cf. [P3]).

Free products of completely positive maps and spectral sets

The aim of the workshop was to give a panoramic view of the main lines of research on various aspects of growth and expansion in finite groups. The main topics included: diameter bounds for Cayley graphs of alternating and classical groups, with results for generating sets that can be arbitrary, random, or containing special elements; constructions of expander families of finite simple groups, with the use of property (T); and character bounds for finite simple groups, yielding applications to word map problems and random walks on Cayley graphs. A series of smaller topics explored connections and similarities to different problems in group theory.

A Cayley graph of a group $H$ is a finite simple graph $\Gamma$ such that ${\rm Aut}(\Gamma)$ contains a subgroup isomorphic to $H$ acting regularly on $V(\Gamma)$, while a Haar graph of $H$ is a finite simple bipartite graph $\Sigma$ such that ${\rm Aut}(\Sigma)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V(\Sigma)$ and the $H$-orbits are equal to the bipartite sets of $\Sigma$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that the groups $D_6, \, D_8, \, D_{10}$ and $Q_8$ are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs (a group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian). As an application, it is also shown that every non-solvable group has a Haar graph which is not a Cayley graph.

<abstract><p>R. Lyons, R. Pemantle and Y. Peres (Ann. Probab. 24 (4), 1996, 1993–2006) conjectured that for a Cayley graph $ G $ with a growth rate $ \mathrm{gr}(G) > 1 $, the speed of a biased random walk exists and is positive for the biased parameter $ \lambda \in (1, \mathrm{gr}(G)) $. And Gábor Pete (Probability and geometry on groups, Chaper 9, 2024) sheds light on the intricate relationship between the spectral radius of the graph and the speed of the biased random walk. Here, we focus on an example of a Cayley graph, a free product of complete graphs. In this paper, we establish the continuity of the spectral radius of biased random walks with respect to the bias parameter in this class of Cayley graphs. Our method relies on the Kesten-Cheeger-Dodziuk-Mohar theorem and the analysis of generating functions.</p></abstract>

Let G be a group. An element g of G is called conjugacy distinguished or c.d. in G if and only if given any element h of G either h is conjugate to g or there is a homomorphism ξ \xi from G onto a finite group such that ξ ( h ) \xi (h) and ξ ( g ) \xi (g) are not conjugate in ξ ( G ) \xi (G) . Following A. Mostowski, a group G is conjugacy separable or c.s. if and only if every element of G is c.d. in G. In this paper we prove that every element conjugate to a cyclically reduced element of length greater than 1 in the free product of two free groups with a cyclic amalgamated subgroup is c.d. We also prove that a group formed by adding a root of an element to a free group is c.s.

A gap has been found in the proof of Proposition 3 of [3]. Consequently, this result and the consequences drawn in the paper have to be considered as conjectures. In particular, it is not proved that the Atiyah conjecture is stable under extensions with free quotient groups and under free products with free groups..

We address the problem of the construction of quantum walks on Cayley graphs. Our main motivation is the relationship between quantum algorithms and quantum walks. In particular, we discuss the choice of the dimension of the local Hilbert space and consider various classes of graphs on which the structure of quantum walks may differ. We completely characterise quantum walks on free groups and present partial results on more general cases. Some examples are given, including a family of quantum walks on the hypercube involving a Clifford Algebra.

The workshop Permutation groups organised by Robert Guralnick (Southern California), Cheryl Praeger (Western Australia), Jan Saxl (Cambridge) and Katrin Tent (Bielefeld) was held August 5th–11th, 2007. The focus was recent developments in permutation group theory and their influence on, and from, group representation theory, algebraic graph theory and other areas of geometry and combinatorics, algebraic geometry, model theory and infinite symmetrical structures. It was well attended with 50 participants. Especially valuable was the program to support young researchers which gave the opportunity for a considerable number of young mathematicians to participate in the workshop. They benefited from the ‘Oberwolfach experience’ and also enlivened and contributed enormously to the success of the meeting. The workshop had a friendly and vibrant atmosphere, aided by the excellent facilities of the Institute. All participants appreciated the beautiful lecture by John G. Thompson entitled ‘The divisor matrix and {\rm SL}(2,\Z) ’. (In fact, this meeting was the first Oberwolfach workshop in over twenty years that Thompson had attended.) There were 26 other talks, and the program featured in each of the first five sessions at least one talk by a student or young researcher at the beginning of their career. This facilitated interaction between younger and more established researchers. The number of talks was restricted to give plenty of time for discussion and cooperation among the participants. Other communications were presented as posters. Highlights of the workshop addressed both fundamental permutation group theory, and also significant outcomes from applying permutation group theory and methods in combinatorics, model theory and other areas. Some topics arose in more than one context, forming new connections. A common feature of many contributions was application of the finite simple group classification, or new results on the structure of simple groups. As examples, we mention two highlights reported by early career participants. The question of base size of permutation actions is of importance in computational group theory as well as in the study of the graph isomorphism problem. Recent research has thrown much light on the base sizes of actions of almost simple groups in particular. Burness reported on the very recent proof of a conjecture of Cameron and Kantor dating from 1993 concerning non-standard permutation actions of almost simple groups: these are all primitive actions apart from actions of alternating and symmetric groups on subsets or partitions, and of classical groups on subspaces or pairs of complementary subspaces. In all the latter families the minimum base size is unbounded as the group order increases. The somewhat surprising conjecture, now proved, is that the minimum base size of non-standard permutation groups is at most 7, with the unique example where this bound is attained being the Mathieu group M_{24} in its natural action on 24 points. Expander graphs play an important role in computer science and combinatorics, and, for example, are significant in modelling and analysing communication networks. In particular, their diameters grow only logarithmically with the numbers of vertices. This property on diameters was proved in 1989 by Babai, Kantor and Lubotzky to hold for suitable Cayley graphs of simple groups: specifically for each finite nonabelian simple group G , they constructed a valency 7 Cayley graph for G with diameter at most a constant times \log(|G|) . Although their graphs were not expanders, they conjectured that there exists an absolute constant k such that all infinite families of simple groups give rise to families of valency k Cayley graphs that are expanders. Kassabov lectured on amazing progress towards proving this conjecture – it is now proved for all simple groups apart from the Suzuki groups. It was a very happy and successful workshop. Indeed, several of the most distinguished participants commented that this was one of the best Oberwolfach meetings they had attended, with plenty of time for collaboration, and outstanding talks. Extended abstracts of the talks mentioned, as well as all the others, are given below (in the order in which they were presented).

An infinite graph Γ is minor excluded if there is a finite graph that is not a minor of Γ. We prove that minor excluded graphs have finite Assouad–Nagata dimension and study minor exclusion for Cayley graphs of finitely generated groups. Our main results and observations are: (1) minor exclusion is not a group property: it depends on the choice of generating set; (2) a group with one end has a generating set for which the Cayley graph is not minor excluded; (3) there are groups that are not minor excluded for any set of generators, like ℤ3; (4) minor exclusion is preserved under free products; and (5) virtually free groups are minor excluded for any choice of finite generating set.

Cayley-graphs of monoids are investigated under a logical point of view. It is shown that the class of monoids, for which the Cayley-graph has a decidable monadic second-order theory, is closed under free products. This result is derived from a result of Walukiewicz, stating that the decidability of monadic second-order theories is preserved under tree-like unfoldings. Concerning first-order logic, it is shown that the class of monoids, for which the Cayley-graph has a decidable first-order theory, is closed under arbitrary graph products, which generalize both, free and direct products. For the proof of this result, tree-like unfoldings are generalized to so-called factorized unfoldings. It is shown that the decidability of the first-order theory of a structure is preserved by factorized unfoldings. Several additional results concerning factorized unfoldings are shown.

Let Γ be a nonelementary hyperbolic group with a word metric d and ∂Γ its hyperbolic boundary equipped with a visual metric for some parameter . Fix a superexponential symmetric probability μ on Γ whose support generates Γ as a semigroup, and denote by ρ the spectral radius of the random walk Y on Γ with step distribution μ. Let ν be a probability on with mean . Let be the branching random walk on Γ with offspring distribution ν and base motion Y , and let be the volume growth rate for the trace of . We prove for that the Hausdorff dimension of the limit set Λ , which is the random subset of consisting of all accumulation points of the trace of , is given by . Furthermore, we prove that is almost surely a deterministic, strictly increasing, and continuous function of , is bounded by the square root of the volume growth rate of Γ , and has critical exponent 1/2 at in the sense that for some positive constant C. We conjecture that the Hausdorff dimension of Λ in the critical case is almost surely. This has been confirmed on free groups or the free product (by amalgamation) of finitely many finite groups equipped with the word metric d defined by the standard generating set. © 2022 Wiley Periodicals LLC.

We study the computational complexity of the Word Problem (WP) in free solvable groups S r , d S_{r,d} , where r ≥ 2 r \geq 2 is the rank and d ≥ 2 d \geq 2 is the solvability class of the group. It is known that the Magnus embedding of S r , d S_{r,d} into matrices provides a polynomial time decision algorithm for WP in a fixed group S r , d S_{r,d} . Unfortunately, the degree of the polynomial grows together with d d , so the uniform algorithm is not polynomial in d d . In this paper we show that WP has time complexity O ( r n log 2 ⁡ n ) O(r n \log _2 n) in S r , 2 S_{r,2} , and O ( n 3 r d ) O(n^3 r d) in S r , d S_{r,d} for d ≥ 3 d \geq 3 . However, it turns out, that a seemingly close problem of computing the geodesic length of elements in S r , 2 S_{r,2} is N P NP -complete. We prove also that one can compute Fox derivatives of elements from S r , d S_{r,d} in time O ( n 3 r d ) O(n^3 r d) ; in particular, one can use efficiently the Magnus embedding in computations with free solvable groups. Our approach is based on such classical tools as the Magnus embedding and Fox calculus, as well as on relatively new geometric ideas; in particular, we establish a direct link between Fox derivatives and geometric flows on Cayley graphs.

We prove that a countable group G can be embedded into a two-generator simple group S which is an amalgamated free product of groups G * F and F, where F and F are free groups on two generators.S is also the product of two commuting free subgroups.If G has solvable word problem, then we can construct a recursive presentation for S.1. Introduction.It is well known that every countable group is a subgroup of a simple two-generator group (see [3] and [7]).The main result of our paper is Theorem A. (a) A countable group G can be embedded into a simple two-generator group S which is an amalgamated free product of G * Fx and F, where F and Fx are both free groups on two generators.S is also the product of two commuting free subgroups.(b) If G has solvable word problem, we can construct the group S of (a) with a recursive presentation.It was first shown by R. J. Thompson [8] by a construction completely different from ours that a countable group with solvable word problem is a subgroup of a recursively presented finitely generated simple group.Our construction of the group S is similar to the construction of the simple free products of R. Camm [1], or see [6] for a slightly simplified version.2. The construction of the group S. Let 77 be a countable infinite group with an element x of order at least 5, and assume that the elements of 77 are numbered by the integers, 77 = {g" i G Z}, such that g0 = 1, gx = x, g2 = x2, g3 = x3, g4 = x4.Let A = (a) and F = (b,c) be free groups on a and b, c, respectively.Then the subgroup U of A * H generated by the elements of the form ", = aW *'eZ-{0}, where p is a permutation of Z -{0}, is a free group freely generated by the elements u" / g Z -{0}, with transversal AinA*H.Similarly, the elements Vj = bjcu\ 7GZ-{0), of F, where a is a second permutation of Z -{0}, freely generate a free subgroup V of F with transversal B = (b).