On the prehistory of growth of groups

The subject of growth of groups has been active in the former Soviet Union since the early 50s and in the West since 1968, when articles of v{S}varc and Milnor have been published, independently. The purpose of this note is to quote a few articles showing that, before 1968 and at least retrospectively, growth has already played some role in various subjects.

Unitary representations of groups, duals, and characters

This is an expository book on unitary representations of topological groups, and of several dual spaces, which are spaces of such representations up to some equivalence. The most important notions are defined for topological groups, but a special attention is paid to the case of discrete groups. The unitary dual of a group $G$ is the space of equivalence classes of its irreducible unitary representations; it is both a topological space and a Borel space. The primitive dual is the space of weak equivalence classes of unitary irreducible representations. The normal quasi-dual is the space of quasi-equivalence classes of traceable factor representations; it is parametrized by characters, which can be finite or infinite. The theory is systematically illustrated by a series of specific examples: Heisenberg groups, affine groups of infinite fields, solvable Baumslag-Solitar groups, lamplighter groups, and general linear groups. Operator algebras play an important role in the exposition, in particular the von Neumann algebras associated to a unitary representation and C*-algebras associated to a locally compact group.

Groups with irreducibly unfaithful subsets for unitary representations

Let $G$ be a group. A subset $F subset G$ is called irreducibly faithful if there exists an irreducible unitary representation $pi$ of $G$ such that $pi(x) eq mathrm{id}$ for all $x in F smallsetminus {e}$. Otherwise $F$ is called irreducibly unfaithful. Given a positive integer $n$, we say that $G$ has Property $P(n)$ if every subset of size $n$ is irreducibly faithful. Every group has $P(1)$, by a classical result of Gelfand and Raikov. Walter proved that every group has $P(2)$. It is easy to see that some groups do not have $P(3)$. We provide a complete description of the irreducibly unfaithful subsets of size $n$ in a countable group $G$ (finite or infinite) with Property $P(n-1)$: it turns out that such a subset is contained in a finite elementary abelian normal subgroup of $G$ of a particular kind. We deduce a characterization of Property $P(n)$ purely in terms of the group structure. It follows that, if a countable group $G$ has $P(n-1)$ and does not have $P(n)$, then $n$ is the cardinality of a projective space over a finite field. A group $G$ has Property $Q(n)$ if, for every subset $F subset G$ of size at most $n$, there exists an irreducible unitary representation $pi$ of $G$ such that $pi(x) e pi(y)$ for any distinct $x, y$ in $F$. Every group has $Q(2)$. For countable groups, it is shown that Property $Q(3)$ is equivalent to $P(3)$, Property $Q(4)$ to $P(6)$, and Property $Q(5)$ to $P(9)$. For $m, n ge 4$, the relation between Properties $P(m)$ and $Q(n)$ is closely related to a well-documented open problem in additive combinatorics.

Brouwer degree, domination of manifolds, and groups presentable by products

For oriented connected closed manifolds of the same dimension, there is a transitive relation: $M$ dominates $N$, or $M ge N$, if there exists a continuous map of non-zero degree from $M$ onto $N$. Section 1 is a reminder on the notion of degree (Brouwer, Hopf), Section 2 shows examples of domination and a first set of obstructions to domination due to Hopf, and Section 3 describes obstructions in terms of Gromovs simplicial volume. In Section 4 we address the particular question of when a given manifold can (or cannot) be dominated by a product. These considerations suggest a notion for groups (fundamental groups), due to D. Kotschick and C. Loh: a group is presentable by a product if it contains two infinite commuting subgroups which generate a subgroup of finite index. The last section shows a small sample of groups which are not presentable by products; examples include appropriate Coxeter groups.

Zooming in on the large-scale geometry of locally compact groups

The purpose of this survey is to describe how locally compact groups can be studied as geometric objects. We will emphasize the main ideas and skip or just sketch most proofs, often referring the reader to our much more detailed book arXiv:1403.3796

Conjugacy growth series of some infinitely generated groups

It is observed that the conjugacy growth series of the infinite fini-tary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed, for other generating sets, for restricted permutational wreath products of finite groups by the finitary symmetric group, and for alternating groups. Similar methods are used to compute usual growth polynomials and conjugacy growth polynomials for finite symmetric groups and alternating groups, with respect to various generating sets of transpositions. Computations suggest a class of finite graphs, that we call partition-complete, which generalizes the class of semi-hamiltonian graphs, and which is of independent interest. The coefficients of a series related to the finitary alternating group satisfy congruence relations analogous to Ramanujan congruences for the partition function. They follow from partly conjectural generalized Ramanujan congruences, as we call them, for which we give numerical evidence in Appendix C.

Amenability and ergodic properties of topological groups: from Bogolyubov onwards

The purpose of this expository article is to revisit the notions of amenability and ergodicity, and to point out that they appear for topological groups that are not necessarily locally compact in articles by Bogolyubov (1939), Fomin (1950), Dixmier (1950), and Rickert (1967).

Metric geometry of locally compact groups

This book offers to study locally compact groups from the point of view of appropriate metrics that can be defined on them, in other words to study Infinite groups as geometric objects, as Gromov writes it in the title of a famous article. The theme has often been restricted to finitely generated groups, but it can favorably be played for locally compact groups. The development of the theory is illustrated by numerous examples, including matrix groups with entries in the the field of real or complex numbers, or other locally compact fields such as p-adic fields, isometry groups of various metric spaces, and, last but not least, discrete group themselves. Word metrics for compactly generated groups play a major role. In the particular case of finitely generated groups, they were introduced by Dehn around 1910 in connection with the Word Problem. Some of the results exposed concern general locally compact groups, such as criteria for the existence of compatible metrics on locally compact groups. Other results concern special classes of groups, for example those mapping onto the group of integers (the Bieri-Strebel splitting theorem for locally compact groups). Prior to their applications to groups, the basic notions of coarse and large-scale geometry are developed in the general framework of metric spaces. Coarse geometry is that part of geometry concerning properties of metric spaces that can be formulated in terms of large distances only. In particular coarse connectedness, coarse simple connectedness, metric coarse equivalences, and quasi-isometries of metric spaces are given special attention. The final chapters are devoted to the more restricted class of compactly presented groups, generalizing finitely presented groups to the locally compact setting. They can indeed be characterized as those compactly generated locally compact groups that are coarsely simply connected.

Amenable groups without finitely presented amenable covers

The goal of this article is to study results and examples concerning finitely presented covers of finitely generated amenable groups. We collect examples of groups $G$ with the following properties: (i) $G$ is finitely generated, (ii) $G$ is amenable, e.g. of intermediate growth, (iii) any finitely presented group $E$ with a quotient isomorphic to $G$ contains non-abelian free subgroups, or the stronger (iii) any finitely presented group with a quotient isomorphic to $G$ is large.

The Fuglede-Kadison determinant, theme and variations

We review the definition of determinants for finite von Neumann algebras, due to Fuglede and Kadison (1952), and a generalisation for appropriate groups of invertible elements in Banach algebras, from a paper by Skandalis and the author (1984). After some reminder on K-theory and Whitehead torsion, we hint at the relevance of these determinants to the study of $L^2$-torsion in topology.