To every dynamical system $(X,\varphi )$ over a totally disconnected compact space, we associate a left-orderable group $T(\varphi )$. It is defined as a group of homeomorphisms of the suspension of $(X,\varphi )$ which preserve every orbit of the suspension flow and act by dyadic piecewise linear homeomorphisms in the flow direction. We show that if the system is minimal, the group is simple and, if it is a subshift, then the group is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple left-orderable groups. We show that if the system is minimal, every action of the corresponding group on the circle has a fixed point. These constitute the first examples of finitely generated left-orderable groups with this fixed point property. We show that for every system $(X,\varphi )$, the group $T(\varphi )$ does not have infinite subgroups with Kazhdan's property $(T)$. In addition, we show that for every minimal subshift, the corresponding group is never finitely presentable. Finally, if $(X,\varphi )$ has a dense orbit, then the isomorphism type of the group $T(\varphi )$ is a complete invariant of flow equivalence of the pair $\{\varphi , \varphi ^{-1}\}$.

We calculate the Bieri–Neumann–Strebel–Renz invariant Σ1(G) for finitely presented residually free groups G and show that its complement in the character sphere S(G) is a finite union of finite intersections of closed sub-spheres in S(G). Furthermore, we find some restrictions on the higher-dimensional homological invariants Σn(G, ℤ) and show for the discrete points Σ2(G)dis, Σ2(G, ℤ)dis and Σ2(G, ℚ)dis in Σ2(G), Σ2(G, ℤ) and Σ2(G, ℚ) that we have the equality Σ2(G)dis = Σ2(G, ℤ)dis = Σ2(G, ℚ)dis.

In this paper we study a notion of HL-extension (HL standing for Herwig–Lascar) for a structure in a finite relational language $\mathcal {L}$. We give a description of all finite minimal HL-extensions of a given finite $\mathcal {L}$-structure. In addition, we study a group-theoretic property considered by Herwig–Lascar and show that it is closed under taking free products. We also introduce notions of coherent extensions and ultraextensive $\mathcal {L}$-structures and show that every countable $\mathcal {L}$-structure can be extended to a countable ultraextensive structure. Finally, it follows from our results that the automorphism group of any countable ultraextensive $\mathcal {L}$-structure has a dense locally finite subgroup.

Isaacs and Seitz conjectured that the derived length of a finite solvable group $G$ is bounded by the cardinality of the set of all irreducible character degrees of $G$. We prove that the conjecture holds for $G$ if the degrees of nonlinear monolithic characters of $G$ having the same kernels are distinct. Also, we show that the conjecture is true when $G$ has at most three nonlinear monolithic characters. We give some sufficient conditions for the inequality related to monolithic characters or real-valued irreducible characters of $G$ when the commutator subgroup of $G$ is supersolvable.

We show that the only summands of the theta divisor on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the powers of the curve and the Fano surface of lines on the threefold. The proof only uses the decomposition theorem for perverse sheaves, some representation theory and the notion of characteristic cycles.

We prove Bogolyubov–Ruzsa-type results for finite subsets of groups with small tripling, |A3| ≤ O(|A|), or small alternation, |AA−1A| ≤ O(|A|). As applications, we obtain a qualitative analogue of Bogolyubov’s lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularity lemma for sets of bounded VC-dimension in finite groups of bounded exponent. The latter result generalizes the abelian case, due to Alon, Fox and Zhao, and gives a quantitative version of previous work of the author, Pillay and Terry.

For a finite group $G$, let $d(G)$ denote the minimal number of elements required to generate $G$. In this paper, we prove sharp upper bounds on $d(H)$ whenever $H$ is a maximal subgroup of a finite almost simple group. In particular, we show that $d(H)\leqslant 5$ and that $d(H)\geqslant 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.

We consider the sequence of powers of a positive definite function on a discrete group. Taking inspiration from random walks on compact quantum groups, we give several examples of situations where a cut-off phenomenon occurs for this sequence, including free groups and infinite Coxeter groups. We also give examples of absence of cut-off using free groups again.

Diagrammatic reducibility DR and its generalization, vertex asphericity VA, are combinatorial tools developed for detecting asphericity of a 2-complex. Here we present tests for a relative version of VA that apply to pairs of 2-complexes $(L,K)$, where K is a subcomplex of L. We show that a relative weight test holds for injective labeled oriented trees, implying that they are VA and hence aspherical. This strengthens a result obtained by the authors in 2017 and simplifies the original proof.

Let $\unicode[STIX]{x1D6E4}$ denote the mapping class group of the plane minus a Cantor set. We show that every action of $\unicode[STIX]{x1D6E4}$ on the circle is either trivial or semiconjugate to a unique minimal action on the so-called simple circle.

In this article, we will prove a full topological version of Popa’s measurable cocycle superrigidity theorem for full shifts [Popa, Cocycle and orbit equivalence superrigidity for malleable actions of $w$-rigid groups. Invent. Math. 170(2) (2007), 243–295]. Let $G$ be a finitely generated group that has one end, undistorted elements and sub-exponential divergence function. Let $H$ be a target group that is complete and admits a compatible bi-invariant metric. Then, every Hölder continuous cocycle for the full shifts of $G$ with value in $H$ is cohomologous to a group homomorphism via a Hölder continuous transfer map. Using the ideas of Behrstock, Druţu, Mosher, Mozes and Sapir [Divergence, thick groups, and short conjugators. Illinois J. Math. 58(4) (2014), 939–980; Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity. Math. Ann. 344(3) (2009), 543–595; Divergence in lattices in semisimple Lie groups and graphs of groups. Trans. Amer. Math. Soc. 362(5) (2010), 2451–2505; Tree-graded spaces and asymptotic cones of groups. Topology 44(5) (2005), 959–1058], we show that the class of our acting groups is large including wide groups having undistorted elements and one-ended groups with strong thick of finite orders. As a consequence, irreducible uniform lattices of most of higher rank connected semisimple Lie groups, mapping class groups of $g$-genus surfaces with $p$-punches, $g\geq 2,p\geq 0$; Richard Thompson groups $F,T,V$; $\text{Aut}(F_{n})$, $\text{Out}(F_{n})$, $n\geq 3$; certain (two-dimensional) Coxeter groups; and one-ended right-angled Artin groups are in our class. This partially extends the main result in Chung and Jiang [Continuous cocycle superrigidity for shifts and groups with one end. Math. Ann. 368(3–4) (2017), 1109–1132].

Steinberg’s tensor product theorem shows that for semisimple algebraic groups, the study of irreducible representations of higher Frobenius kernels reduces to the study of irreducible representations of the first Frobenius kernel. In the preceding paper in this series, deforming the distribution algebra of a higher Frobenius kernel yielded a family of deformations called higher reduced enveloping algebras. In this paper, we prove that the Steinberg decomposition can be similarly deformed, allowing us to reduce representation theoretic questions about these algebras to questions about reduced enveloping algebras. We use this to derive structural results about modules over these algebras. Separately, we also show that many of the results in the preceding paper hold without an assumption of reductivity.

We present three examples of countable homogeneous structures (also called Fraïssé limits) whose automorphism groups are not universal, namely, fail to contain isomorphic copies of all automorphism groups of their substructures.

Our first example is a particular case of a rather general construction on Fraïssé classes, which we call diversification, leading to automorphism groups containing copies of all finite groups. Our second example is a special case of another general construction on Fraïssé classes, the mixed sums, leading to a Fraïssé class with all finite symmetric groups appearing as automorphism groups and at the same time with a torsion-free automorphism group of its Fraïssé limit. Our last example is a Fraïssé class of finite models with arbitrarily large finite abelian automorphism groups, such that the automorphism group of its Fraïssé limit is again torsion-free.

It is known that there exists a first-order sentence that holds in a finite group if and only if the group is soluble. Here it is shown that the corresponding statements with ‘solubility’ replaced by ‘nilpotence’ and ‘perfectness’, among others, are false.

These facts present difficulties for the study of pseudofinite groups. However, a very weak form of Frattini’s theorem on the nilpotence of the Frattini subgroup of a finite group is proved for pseudofinite groups.

For a finite group $G$, define $l(G)=(\prod _{g\in G}o(g))^{1/|G|}/|G|$, where $o(g)$ denotes the order of $g\in G$. We prove that if $l(G)>l(A_{5}),l(G)>l(A_{4}),l(G)>l(S_{3}),l(G)>l(Q_{8})$ or $l(G)>l(C_{2}\times C_{2})$, then $G$ is solvable, supersolvable, nilpotent, abelian or cyclic, respectively.

For a reductive group $G$ over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group $H$ with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on $G$ with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group $H$, after passing to asymptotic versions. The other is a similar relationship between representations of $G(\mathbb{F}_{q})$ with a fixed semisimple parameter and unipotent representations of $H(\mathbb{F}_{q})$.

In their book Subgroup Growth, Lubotzky and Segal asked: What are the possible types of subgroup growth of the pro-$p$ group? In this paper, we construct certain extensions of the Grigorchuk group and the Gupta–Sidki groups, which have all possible types of subgroup growth between $n^{(\log n)^{2}}$ and $e^{n}$. Thus, we give an almost complete answer to Lubotzky and Segal’s question. In addition, we show that a class of pro-$p$ branch groups, including the Grigorchuk group and the Gupta–Sidki groups, all have subgroup growth type $n^{\log n}$.

We study a relative variant of Serre’s notion of $G$-complete reducibility for a reductive algebraic group $G$. We let $K$ be a reductive subgroup of $G$, and consider subgroups of $G$ that normalize the identity component $K^{\circ }$. We show that such a subgroup is relatively $G$-completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$, as well as ‘rational’ versions over nonalgebraically closed fields.

We observe an inductive structure in a large class of Artin groups of finite real, complex and affine types and exploit this information to deduce the Farrell–Jones isomorphism conjecture for these groups.

Motivated by the Bruhat and Cartan decompositions of general linear groups over local fields, we enumerate double cosets of the group of label-preserving automorphisms of a label-regular tree over the fixator of an end of the tree and over maximal compact open subgroups. This enumeration is used to show that every continuous homomorphism from the automorphism group of a label-regular tree has closed range.