On isomorphisms to a free group and beyond

Subset currents on hyperbolic groups were introduced by Kapovich and Nagnibeda as a generalization of geodesic currents on hyperbolic groups, which were introduced by Bonahon and have been successfully studied in the case of the fundamental group π 1 ( Σ ) \pi _1 (\Sigma ) of a compact hyperbolic surface Σ \Sigma . Kapovich and Nagnibeda particularly studied subset currents on free groups. In this article, we develop the theory of subset currents on π 1 ( Σ ) \pi _1(\Sigma ) , which we call subset currents on Σ \Sigma . We prove that the space S C ( Σ ) \mathrm {SC}(\Sigma ) of subset currents on Σ \Sigma is a measure-theoretic completion of the set of conjugacy classes of non-trivial finitely generated subgroups of π 1 ( Σ ) \pi _1 (\Sigma ) , each of which geometrically corresponds to a convex core of a covering space of Σ \Sigma . This result was proved by Kapovich-Nagnibeda in the case of free groups, and is also a generalization of Bonahon’s result on geodesic currents on hyperbolic groups. We will also generalize several other results of them. Especially, we extend the (geometric) intersection number of two closed geodesics on Σ \Sigma to the intersection number of two convex cores on Σ \Sigma and, in addition, to a continuous R ≥ 0 \mathbb {R}_{\geq 0} -bilinear functional on S C ( Σ ) \mathrm {SC}(\Sigma ) .

Brian Hartley asked me whether a free (nilpotent of class 2 and exponent p2)-group of countable rank has a faithful linear representation of finite degree, p here being a prime of course. The answer is yes. The point is that this then yields via work of F. Leinen and M. J. Tomkinson, see [3,3.6] an image of a linear p-group, which is not even finitary linear. The question of which relatively free groups have faithful linear representations dates back at least to work of W. Magnus in the 1930's, see [4, pp. 33, 34 and the final comment on p. 40] for a discussion of this. Our construction, which works more generally, is a further contribution. We write ℜc for the variety of nilpotent groups of class at most c and (ℭ9 for the variety of groups of exponent dividing q.

There is an important class of finitely presented groups known as automatic groups, in which many computations can be undertaken efficiently. In particular, there is a normal form for group elements which is the language of a finite state automation, and arbitrary words in the group generators can be reduced to normal form in quadratic time. The normal form can also be used to enumerate group elements, and to compute the (rational) growth function of the group. The theory of automatic groups is developed [2], and the algorithms are described in more detail in [4]. Implementations of the algorithms are available in a software package of the author ([3]).The word-hyperbolic groups form an important subclass of the automatic groups. They are defined by the property that geodesic triangles in the Cayley graph of the group are uniformly thin. In other words, there is a constant e, such a that any point on one of the sides of such a triangle is within distance e of the union of the other two sides of the triangle. Small cancellation groups and groups acting discretely and cocompactly on hyperbolic space are examples of word-hyperbolic groups. Their theory is developed in [1].The word problem in a word-hyperbolic group can be solved by a Dehn algorithm in linear time. Furthermore, Epstein has shown that the conjugacy problem can be solved in time O(n ln n). However, both of these algorithms require a prior knowledge of the constant e, and it does not seem to be easy to compute this constant in general.In practice, the quadratic time algorithm for the word problem coming from the automatic groups procedures performs very effectively, and it would be useful to have alternatives for the conjugacy problem.The automatic groups machinery does provide such an algorithm (which works in the the intermediate class of biautomatic groups), but that unfortunately is exponential.Alternative algorithms for conjugacy testing of elements and subgroups are currently being developed and implemented by a research student of mine, Joe Marshall. They have the minor disadvanatage that their basic versions work only in torsion-free groups, but adaptations to the more general case seem possible. They include also a test for malnormality of a subgroup H of G, which means that g-1Hg∩H = 1 for all g ∈ G\H. This test is important in topological applications.They make essential use of the boundary of the group. The points on the boundary can be defined as the equivalence classes of infinite geodesic rays starting from the base point of the Cayley graph, where two such rays are equivalent if they remain a bounded distance apart. The precise bound involved here turns out to be another universal constant d of the Cayley graph, but unlike e it can be computed in practice relatively easily. Roughly speaking, a certain linear function of d provides us with a bound on the length of the elements g that we need to consider as possible conjugating elements when testing for conjugacy or malnormality. To make the algorithm practical, we need to restrict this set of potential conjugators considerably more than this, and Marshall has come up with various tricks for doing this.

Lyndon's group FZ[x] is the free exponential group over the ring of integral polynomials Z[x]. This group, introduced by Lyndon in the 1960s, continues to be of interest to group theorists due to its importance in the study of first-order properties of free groups, in particular, equations over free groups. One of the crucial results of Lyndon's study was that the group FZ[x] is fully residually F; i.e. for any finite collection of nontrivial elements in FZ[x] there exists a homomorphism φ : FZ[x] → F which is the identity on F and maps the given elements of FZ[x] into nontrivial elements of F. The importance of FZ[x] was further emphasized when Kharlampovich and Myasnikov proved in [3] that a finitely generated group is fully residually free if and only if it is embeddable into FZ[x]. Lyndon's group and its subgroups play a vital role in the technique employed by O. Kharlampovich and A. Myasnikov in their solution of the famous Tarski problem on the decidability of the elementary theory of a free group (see [4, 5]). In this paper, we show that Lyndon's group is conjugately residually free, i.e. it is possible to map FZ[x] to the free group F preserving the nonconjugacy of two elements. This result is a further step towards the understanding of the properties of FZ[x]; moreover, it is closely related to the problem of "lifting solutions" of equations from F to FZ[x], since our result implies that the solutions can indeed be "lifted" from F to FZ[x] for equations of the type x-1 c1 x = c2. The structure of Lyndon's group, described by A. Myasnikov and V. Remeslennikov in [8], involves an infinite sequence of free constructions of a specific type, called free extensions of centralizers. For more results on residual properties of certain types of free constructions, see also the works of Ribes, Segal and Zalesskii (for example [9]).

This meeting was focused on asymptotic aspects of group theory. The resulting problems lead, in particular, to the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients. Also, properties of infinite families of finite groups are at the center of interest in the field. We had talks on the most important recent developments in the field of asymptotic and profinite group theory. We single out some of these new results. The classification of finite simple groups has provided an impetus for the detailed investigation of properties of families of groups of Lie type and of alternating groups. This has produced many results of asymptotic or probabilistic nature, describing in a quantitative way properties that hold for all sufficiently large simple groups, or else that hold ‘almost surely’ as the orders of the groups are tending to infinity. Here we had a talk of Aner Shalev on his recent solution (with coauthors) of Ore's conjecture. Recently there has been much activity in the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients, or their finite-index subgroups. The theory of subgroup growth is now well established; the theory of the associated zeta functions is making good progress. More subtle variations of subgroup growth are also developing into significant theories in their own right: there are important new results on representation growth – of arithmetic groups, nilpotent groups and pro-p groups; and on maximal subgroup growth, with applications to probabilistic generation properties. Profinite groups are ‘asymptotic limits’ of finite groups. The nature of verbal subgroups and verbal mappings, in both finite and profinite groups, has been the focus of much recent work. This has led among other things to the solution of Serre's problem on subgroups of finite index in finitely generated profinite groups; other significant results include the characterisation of closed verbal subgroups in pro-p groups and new characterisations of solubility in finite groups. Branch groups, both discrete and profinite, are certain groups of automorphisms of rooted trees; introduced by Grigorchuk and others over 20 years ago, they have provided the solution to several outstanding problems, and their study continues to provide new results. Some of them were reported on in our meeting. We also had talks from important fields of mathematics in which profinite and asymptotic group theory finds application. There is, for instance, the study of the absolute Galois group of the rationals and in particular its closed pro-p subgroups. Methods from pro-p group theory have successfully been applied in order to understand fundamental groups of (hyperbolic) 3-manifolds. The interplay between the topological fundamental group of a complex algebraic surface and its profinite completion which is the etale fundamental group of the surface is the subject of recent research. We believe that the meeting was exciting and successful. The high quality lectures covered a lot of very recent results in the field. There were many intense discussions between mathematicians of different fields. Altogether there were over 50 participants from all over the world. The percentage of young researchers was very high. The following extended abstracts were collected and edited by Dr. Evija Ribnere.

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.

We shall characterize the complete first order theories of torsion free abelian groups having algebraically prime models, elementarily prime models, minimal models, etc., using the well known invariants from [9], [3]. This study was suggested by the questions raised in [2]. Group here will always mean abelian group. For reference on groups see Fuchs [5], [6], for model theory Chang-Keisler [1]. All theories considered are theories in the first order language of group theory. A group G is minimal (strictly minimal) if H < G implies H = G (H-= G), and elementarily prime (algebraically prime) if G is elementarily (algebraically) embeddable in every H G. p, q always denote prime numbers, P the set of all primes. The p-rank of a group G, p-rk(G), is the dimension of G/pG as a vector space over Z/pZ. We define tf(p, G): = min {p-rk(G), ~o}. By the results of [9], [3], two nontrivial torsion free abelian groups G and H are elementarily equivalent iff for all primes p tf(p, G) = tf(p, H). This gives a natural 1-1 correspondence between the complete theories of nontrivial torsion free abelian groups and the functions /3 : P --~ ~o + 1. /3 will always stand for such a function, and the corresponding complete theory will be denoted by T 0. S o is the set of primes p for which/3(p) ~ 0. For more definitions see section 3. Let 13 be a function from P to o) + 1. We shall study the following questions: Does T o have an elementarily (algebraically) prime model? Does T o have a minimal (strictly minimal) model, and if this is the case, how many such models does T o have? Do the minimal and strictly minimal models of T o coincide? These questions have been studied for the special case of the theory of the

We define quasiconvexity cone Qcone(τ) over an infinite hyperbolic (in the sense of Gromov) group τ as the set of conjugacy classes of infinite quasiconvex subgroups H⊂τ and show that the abelian group of Qcone(τ)-divisors, i.e. finite sums of points from Qcone(τ) with integer coefficients, can be equipped with a natural structure of commutative associative ring with identity. Euler characteristic can be considered as a rational-valued function on Qcone(τ). This approach gives another point of view on the strengthened form of Hanna Neumann's conjecture on the maximal rank of the intersection of two finitely generated subgroups of the free group on two generators.

We introduce the notion of graphical discreteness to group theory. A finitely generated group is graphically discrete if whenever it acts geometrically on a locally finite graph, the automorphism group of the graph is compact‐by‐discrete. Notable examples include finitely generated nilpotent groups, most lattices in semisimple Lie groups, and irreducible nongeometric 3‐manifold groups. We show graphs of groups with graphically discrete vertex groups frequently have strong rigidity properties. We prove free products of one‐ended virtually torsion‐free graphically discrete groups are action rigid within the class of virtually torsion‐free groups. We also prove quasi‐isometric rigidity for many hyperbolic graphs of groups whose vertex groups are closed hyperbolic manifold groups and whose edge groups are nonelementary quasi‐convex subgroups. This includes the case of two hyperbolic 3‐manifold groups amalgamated along a quasi‐convex malnormal non‐abelian free subgroup. We provide several additional examples of graphically discrete groups and illustrate this property is not a commensurability invariant.

Major progress in group theory occurred in the nineteenth century but the evolution of group theory began already in the latter part of 18th century. Some characteristic features of 19th century mathematics which had crucial impact on this evolution are concern for rigor and abstraction and the view that mathematics is a human activity without necessarily referring to physical situations. In 1770, Joseph Louis Lagrange (1736-1813) wrote his seminal memoir Reflections on the solution of algebraic equations. He considered ‘abstract’ questions like whether every equation has a root and, if so, how many were real/complex/positive/negative? The problem of algebraically solving 5th degree equations was a major preoccupation (right from the 17th century) to which Lagrange lent his major efforts in this paper. His beautiful idea (now going under the name of Lagrange’s resolvent) is to ‘reduce’ a general equation to auxiliary (resolvent) equations which have one degree less. Later, the theory of finite abelian groups evolved from Carl Friedrich Gauss’s famous “Disquisitiones Arithmeticae”. Gauss (1777-1855) established many of the important properties though he did not use the terminology of group theory. In his work, finite abelian groups4 Math Unlimitedappeared in different forms like the additive group Zn of integers modulo n, the multiplicative group Z∗n of integers modulo n relatively prime to n, the group of equivalence classes of binary quadratic forms, and the group of n-th roots of unity. In 1872, Felix Klein delivered a famous lecture A Comparative Review of Recent Researches in Geometry. The aim of his (so-called) Erlangen Program was the classification of geometry as the study of invariants under various groups of transformations. So, the groups appear here “geometrically” as groups of rigid motions, of similarities, or as the hyperbolic group etc. During the analysis of the connections between the different geometries, the focus was on the study of properties of figures invariant under transformations. Soon after, the focus shifted to a study of the transformations themselves. Thus the study of the geometric relations between figures got converted to the study of the associated transformations. In 1874, Sophus Lie introduced his “theory of continuous transformation groups” what we basically call Lie groups today. Poincaré and Klein began their work on the so-called automorphic functions and the groups associated with them around 1876. Automorphic functions are generalizations of the circular, hyperbolic, elliptic, and other functions of elementary analysis. They are functions of a complex variable z, analytic in some domain, and invariant under the group of transformations x′ = ax+b(a, b, c, d real or complex and ad − bc 0), or under some subgroup of this group. We end this introduction with the ancient problem of finding all those positive integers (called congruent numbers) which are areas of right-angled triangles with rational sides. To this date, the general result describing all of them precisely is not proved. However, the problem can be re-stated in terms of certain groups called elliptic curves and this helps in getting a hold on the problem and in obtaining several partial results. Indeed, if n is a positive integer, then the rational numbers x, y satisfying y2 = x3 − n2x form a group law which can be described geometrically. Then, n is a congruent number if and only if, this group is infinite!

The question of studying the property of antinormality of subgroups is important, sincethis property is used in the study of various properties of groups (for example, hyperbolicity of groups) and various algorithmic problems in groups.The proposed article gives a complete description of all antinormal two-generated subgroups with a certain condition imposed on the generators of this subgroup contained in a free group of rank greater than one.

Combinatorial group theory concerns itself with the theory of group presentations, that is, it deals with groups specified by a set of generators and corresponding defining relations. The theory has its origins and roots in topology and complex analysis and in particular in the theory of the fundamental groups of combinatorial cell complexes. Because of its nature and origins, combinatorial group theory comes into contact with, and uses, many different areas of mathematics: clearly algebra and topology as mentioned above, but also hyperbolic geometry via the study of the Cayley graph; pure mathematical logic through the study of various decision problems; and computer science through the study of computational aspects of group theory and rewriting systems. These last three areas have been the subject of increased interest and intense study during the 1990s. The ties with hyperbolic geometry have evolved into geometric group theory—especially the theory of the hyperbolic groups—by considering the groups themselves as geometric objects via the Cayley graph. The ties with computation and computer science have evolved into the theory of automatic groups, where rewriting and computability properties are paramount. Finally, there have been extensive studies on the nature of solutions of the various decision problems and on the logical underpinnings of the the whole theory. Relative to the latter there has been a great deal of work on the Tarski conjectures (see below) and their ties to purely group theoretical properties.KeywordsFree GroupCayley GraphFree ProductHyperbolic GroupMaximal Abelian SubgroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We show that there exists an algorithm to decide any single equation in the Heisenberg group in finite time. The method works for all two-step nilpotent groups with rank-one commutator, which includes the higher Heisenberg groups. We also prove that the decision problem for systems of equations is unsolvable in all non-abelian free nilpotent groups.

The iso-spectrum problem for marked lengnth spectrum for Riemannian manifolds of negative curvature has a rich history. We rephrased the problems for metrics on discrete groups, discussed its connection to a conjecture by Margulis, and proved some results for “total relatively hyperbolic groups” in Koji Fujiwara, Journal of Topology and Analysis, 7(2), 345–359 (2015). This is a note from my talk on that paper and mainly discuss the connection between Riemannian geometry and group theory, and also some questions.

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.