The normal residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients. We show that any S-arithmetic subgroup of a higher rank Chevalley group G has normal residual finiteness growth ndim (G).

We consider finite groups in which, for all primes p, the p-part of the length of any conjugacy class is trivial or fixed. We obtain a full description in the case in which for each prime divisor p of the order of the group there exists a noncentral conjugacy class of p-power size.

In Cossey and Stonehewer [‘On the rarity of quasinormal subgroups’, Rend. Semin. Mat. Univ. Padova125 (2011), 81–105] it is shown that for any odd prime p and integer n≥3, there is a finite p-group G of exponent pn containing a quasinormal subgroup H of exponent pn−1 such that the nontrivial quasinormal subgroups of G lying in H can have exponent only p, pn−1 or, when n≥4 , pn−2. Thus large sections of these groups are devoid of quasinormal subgroups. The authors ask in that paper if there is a nontrivial subgroup-theoretic property 𝔛 of finite p-groups such that (i) 𝔛 is invariant under subgroup lattice isomorphisms and (ii) every chain of 𝔛-subgroups of a finite p-group can be refined to a composition series of 𝔛-subgroups. Failing this, can such a chain always be refined to a series of 𝔛-subgroups in which the intervals between adjacent terms are restricted in some significant way? The present work embarks upon this quest.

Assume that G is a solvable group whose elementary abelian sections are all finite. Suppose, further, that p is a prime such that G fails to contain any subgroups isomorphic to Cp∞. We show that if G is nilpotent, then the pro-p completion map induces an isomorphism for any discrete -module M of finite p-power order. For the general case, we prove that G contains a normal subgroup N of finite index such that the map is an isomorphism for any discrete -module M of finite p-power order. Moreover, if G lacks any Cp∞-sections, the subgroup N enjoys some additional special properties with respect to its pro-p topology.

Groups having exactly one normaliser are well known. They are the Dedekind groups. All finite groups having exactly two normalisers were classified by Pérez-Ramos [‘Groups with two normalizers’, Arch. Math.50 (1988), 199–203], and Camp-Mora [‘Locally finite groups with two normalizers’, Comm. Algebra28 (2000), 5475–5480] generalised that result to locally finite groups. Then Tota [‘Groups with a finite number of normalizer subgroups’, Comm. Algebra32 (2004), 4667–4674] investigated properties (such as solubility) of arbitrary groups with two, three and four normalisers. In this paper we prove that every finite group with at most 20 normalisers is soluble. Also we characterise all nonabelian simple (not necessarily finite) groups with at most 57 normalisers.

In the paper ‘Bruhat–Tits theory from Berkovich's point of view. I. Realizations and compactifications of buildings’, we investigated various realizations of the Bruhat–Tits building of a connected and reductive linear algebraic group G over a non-Archimedean field k in the framework of Berkovich's non-Archimedean analytic geometry. We studied in detail the compactifications of the building which naturally arise from this point of view. In the present paper, we give a representation theoretic flavour to these compactifications, following Satake's original constructions for Riemannian symmetric spaces.

We first prove that Berkovich compactifications of a building coincide with the compactifications, previously introduced by the third named author and obtained by a gluing procedure. Then we show how to recover them from an absolutely irreducible linear representation of G by embedding in the building of the general linear group of the representation space, compactified in a suitable way. Existence of such an embedding is a special case of Landvogt's general results on functoriality of buildings, but we also give another natural construction of an equivariant embedding, which relies decisively on Berkovich geometry.

Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a Kähler group on the real hyperbolic space of dimension at least three factors through a homomorphism onto a cocompact discrete subgroup of PSL2(ℝ). We also study actions of Kähler groups on infinite-dimensional real hyperbolic spaces, describe some exotic actions of PSL2(ℝ) on these spaces, and give an application to the study of the Cremona group.

Let G be a finite p-solvable group. We prove that if G has exactly two conjugacy class sizes of p′-elements of prime power order, say 1 and m, then m=paqb, for two distinct primes p and q, and G either has an abelian p-complement or G=PQ×A, with P and Q a Sylow p-subgroup and a Sylow q-subgroup of G, respectively, and A is abelian. In particular, we provide a new extension of Itô’s theorem on groups having exactly two class sizes for elements of prime power order.

If G is a semisimple Lie group of real rank at least two and Γ is an irreducible lattice in G, then every homomorphism from Γ to the outer automorphism group of a finitely generated free group has finite image.

Let G be a group of odd order that contains a non-central element x whose order is either a prime p ≥ 5 or 3l, with l ≥ 2. Then, in , the group of units of ℤG, we can find an alternating unit u based on x, and another unit v, which can be either a bicyclic or an alternating unit, such that for all sufficiently large integers m we have that 〈um, vm〉 = 〈um〉 ∗ 〈vm〉 ≌ ℤ ∗ ℤ

The following theorem is proved. Let m, k and n be positive integers. There exists a number η=η(m,k,n) depending only on m, k and n such that if G is any residually finite group satisfying the condition that the product of any η commutators of the form [xm,y1,…,yk ]is of order dividing n, then the verbal subgroup of G corresponding to the word w=[xm,y1,…,yk ]is locally finite.

We show that the restriction of the Dehornoy ordering to an appropriate free subgroup of the three-strand braid group defines a left-ordering of the free group on k generators, k>1, that has no convex subgroups.

The computation of growth series for the higher Baumslag–Solitar groups is an open problem first posed by de la Harpe and Grigorchuk. We study the growth of the horocyclic subgroup as the key to the overall growth of these Baumslag–Solitar groups BS(p,q), where 1<p<q. In fact, the overall growth series can be represented as a modified convolution product with one of the factors being based on the series for the horocyclic subgroup. We exhibit two distinct algorithms that compute the growth of the horocyclic subgroup and discuss the time and space complexity of these algorithms. We show that when p divides q, the horocyclic subgroup has a geodesic combing whose words form a context-free (in fact, one-counter) language. A theorem of Chomsky–Schützenberger allows us to compute the growth series for this subgroup, which is rational. When p does not divide q, we show that no geodesic combing for the horocyclic subgroup forms a context-free language, although there is a context-sensitive geodesic combing. We exhibit a specific linearly bounded Turing machine that accepts this language (with quadratic time complexity) in the case of BS(2,3) and outline the Turing machine construction in the general case.

A conjecture of Gromov states that a one-ended word-hyperbolic group must contain a subgroup that is isomorphic to the fundamental group of a closed hyperbolic surface. Recent papers by Gordon and Wilton and by Kim and Wilton give sufficient conditions for hyperbolic surface groups to be embedded in a hyperbolic Baumslag double G. Using Nielsen cancellation methods based on techniques from previous work by the second author, we prove that a hyperbolic orientable surface group of genus 2 is embedded in a hyperbolic Baumslag double if and only if the amalgamated word W is a commutator: that is, W = [U, V] for some elements U, V ∈ F. Furthermore, a hyperbolic Baumslag double G contains a non-orientable surface group of genus 4 if and only if W = X2Y2 for some X, Y ∈ F. G can contain no non-orientable surface group of smaller genus.

Shumyatsky and the second author proved that if G is a finitely generated residually finite p-group satisfying a law, then, for almost all primes p, the fact that a normal and commutator-closed set of generators satisfies a positive law implies that the whole of G also satisfies a (possibly different) positive law. In this paper, we construct a counterexample showing that the hypothesis of finite generation of the group G cannot be dispensed with.

We extend some results known for FC-groups to the class FC* of generalized FC-groups introduced in de Giovanni et al. [‘Groups with restricted conjugacy classes’, Serdica Math. J.28(3) (2002), 241–254]. The main theorems pertain to the join of pronormal subgroups. The relevant role that the Wielandt subgroup plays in an FC*-group is pointed out.

A finite group is called a CH-group if for every x,y∈G∖Z(G), xy=yx implies that . Applying results of Schmidt [‘Zentralisatorverbände endlicher Gruppen’, Rend. Sem. Mat. Univ. Padova44 (1970), 97–131] and Rebmann [‘F-Gruppen’, Arch. Math. 22 (1971), 225–230] concerning CA-groups and F-groups, the structure of CH-groups is determined, up to that of CH-groups of prime-power order. Upper bounds are found for the derived length of nilpotent and solvable CH-groups.

In this paper we construct the maximal subgroups of geometric type of the orthogonal groups in dimension d over GF(q) in O(d3+d2log q+log qlog log q) finite field operations.

We consider finite p-groups G in which every cyclic subgroup has at most p conjugates. We show that the derived subgroup of such a group has order at most p2. Further, if the stronger condition holds that all subgroups have at most p conjugates then the central factor group has order p4 at most.

Let G be a finitely generated group. We investigate the graph ΓM(G), whose vertices are the maximal subgroups of G and where two vertices M1 and M2 are joined by an edge whenever M1∩M2≠1. We show that if G is a finite simple group then the graph ΓM(G) is connected and its diameter is 62 at most. We also show that if G is a finite group, then ΓM(G) either is connected or has at least two vertices and no edges. Finite groups G with a nonconnected graph ΓM(G) are classified. They are all solvable groups, and if G is a finite solvable group with a connected graph ΓM(G), then the diameter of ΓM(G) is at most 2. In the infinite case, we determine the structure of finitely generated infinite nonsimple groups G with a nonconnected graph ΓM(G). In particular, we show that if G is a finitely generated locally graded group with a nonconnected graph ΓM(G), then G must be finite.