Let G be a finite transitive permutation group on $\Omega $. The G-invariant partitions form a sublattice of the lattice of all partitions of $\Omega $, having the further property that all its elements are uniform (that is, have all parts of the same size). If, in addition, all the equivalence relations defining the partitions commute, then the relations form an orthogonal block structure, a concept from statistics; in this case the lattice is modular. If it is distributive, then we have a poset block structure, whose automorphism group is a generalised wreath product. We examine permutation groups with these properties, which we call the OB property and PB property respectively, and in particular investigate when direct and wreath products of groups with these properties also have these properties.

A famous theorem on permutation groups asserts that a transitive imprimitive group G is embeddable in the wreath product of two factors obtained from the group (the group induced on a block by its setwise stabiliser, and the group induced on the set of blocks by G). We extend this theorem to groups with the PB property, embedding them into generalised wreath products. We show that the map from posets to generalised wreath products preserves intersections and inclusions.

We have included background and historical material on these concepts.

We study the hypersimplex under the action of the symmetric group $S_n$ by coordinate permutation. We prove that its equivariant volume, given by the evaluation of its equivariant $H^*$-series at $1$, is the permutation character of decorated ordered set partitions under the natural action of $S_n$. This verifies a conjecture of Stapledon for the hypersimplex. To prove this result, we give a formula for the coefficients of the $H^*$-polynomial. Additionally, for the $(2,n)$-hypersimplex, we use this formula to show that trivial character need not appear as a direct summand of a coefficient of the $H^*$-polynomial, which gives a family of counterexamples to a different conjecture of Stapledon.

We study the restriction of the absolute order on a Coxeter group W to an interval $[1,w]_T$, where $w\in W$ is an involution. We characterise and classify those involutions w for which $[1,w]_T$ is a lattice, using the notion of involutive parabolic subgroups.

We consider the associated graded $\bigoplus_{k\geq 1} \Gamma_k \mathcal{I} /\Gamma_{k+1} \mathcal{I} $ of the lower central series $\mathcal{I}\,=\,\Gamma_1 \mathcal{I}\supset \Gamma_2 \mathcal{I}\supset \Gamma_3 \mathcal{I} \supset \cdots$ of the Torelli group $\mathcal{I}$ of a compact oriented surface. Its degree-one part is well understood by D. Johnson’s seminal works on the abelianization of the Torelli group. The knowledge of the degree-two part $(\Gamma_2 \mathcal{I} / \Gamma_3 \mathcal{I})\otimes \mathbb{Q}$ with rational coefficients arises from works of S. Morita on the Casson invariant and R. Hain on the Malcev completion of $\mathcal{I}$. Here, we prove that the abelian group $\Gamma_2 \mathcal{I} / \Gamma_3 \mathcal{I}$ is torsion-free, and we describe it as a lattice in a rational vector space. As an application, the group $\mathcal{I}/\Gamma_3 \mathcal{I}$ is computed, and it is shown to embed in the group of homology cylinders modulo the surgery relation of $Y_3$-equivalence.

We study an abstract group of reversible Turing machines. In our model, each machine is interpreted as a homeomorphism over a space which represents a tape filled with symbols and a head carrying a state. These homeomorphisms can only modify the tape at a bounded distance around the head, change the state, and move the head in a bounded way. We study three natural subgroups arising in this model: the group of finite-state automata, which generalizes the topological full groups studied in topological dynamics and the theory of orbit-equivalence; the group of oblivious Turing machines whose movement is independent of tape contents, which generalizes lamplighter groups and has connections to the study of universal reversible logical gates, and the group of elementary Turing machines, which are the machines which are obtained by composing finite-state automata and oblivious Turing machines.

We show that both the group of oblivious Turing machines and that of elementary Turing machines are finitely generated, while the group of finite-state automata and the group of reversible Turing machines are not. We show that the group of elementary Turing machines has undecidable torsion problem. From this, we also obtain that the group of cellular automata (more generally, the automorphism group of any uncountable one-dimensional sofic subshift) contains a finitely generated subgroup with undecidable torsion problem. We also show that the torsion problem is undecidable for the topological full group of a full $\mathbb {Z}^d$-shift on a nontrivial alphabet if and only if $d \geq 2$.

We prove that Brinkmann’s problems are decidable for endomorphisms of $F_n\times F_m$: given $(x,y),(z,w)\in F_n\times F_m$ and $\Phi \in \mathrm {End}(F_n\times F_m)$, it is decidable whether there is some $k\in \mathbb {N}$ such that $(x,y)\Phi ^k=(z,w)$ (or $(x,y)\Phi ^k\sim (z,w)$). We also prove decidability of a two-sided version of Brinkmann’s conjugacy problem for injective endomorphisms which, from the work of Logan, yields a solution to the conjugacy problem in ascending HNN-extensions of $F_n\times F_m$. Finally, we study the dynamics of automorphisms of $F_n\times F_m$ at the infinity, proving that that their dynamics at the infinity is asymptotically periodic, as occurs in the free and free-abelian times free cases.

We study new properties of generalised Harish-Chandra theory aiming at explaining the inductive local-global conditions for finite groups of Lie type in nondefining characteristic. In particular, we consider a parametrisation of generalised Harish-Chandra series that is compatible with Clifford theory and with the action of automorphisms on irreducible characters and we reduce it to the verification of certain requirements on stabilisers and extendibility of characters. This parametrisation is used by the author in a separate paper to obtain new conjectures for finite reductive groups that can be seen as geometric realisations of the local-global counting conjectures and their inductive conditions. As a by-product, we extend the parametrisation of generalised Harish-Chandra series given by Broué–Malle–Michel to the nonunipotent case by assuming maximal extendibility.

Given a morphism $\varphi \;:\; G \to A \wr B$ from a finitely presented group G to a wreath product $A \wr B$, we show that, if the image of $\varphi$ is a sufficiently large subgroup, then $\mathrm{ker}(\varphi)$ contains a non-abelian free subgroup and $\varphi$ factors through an acylindrically hyperbolic quotient of G. As direct applications, we classify the finitely presented subgroups in $A \wr B$ up to isomorphism and we deduce that a finitely presented group having a wreath product $(\text{non-trivial}) \wr (\text{infinite})$ as a quotient must be SQ-universal (extending theorems of Baumslag and Cornulier–Kar). Finally, we exploit our theorem in order to describe the structure of the automorphism groups of several families of wreath products, highlighting an interesting connection with the Kaplansky conjecture on units in group rings.

Let $W_{\mathrm {aff}}$ be an extended affine Weyl group, $\mathbf {H}$ be the corresponding affine Hecke algebra over the ring $\mathbb {C}[\mathbf {q}^{\frac {1}{2}}, \mathbf {q}^{-\frac {1}{2}}]$, and J be Lusztig’s asymptotic Hecke algebra, viewed as a based ring with basis $\{t_w\}$. Viewing J as a subalgebra of the $(\mathbf {q}^{-\frac {1}{2}})$-adic completion of $\mathbf {H}$ via Lusztig’s map $\phi $, we use Harish-Chandra’s Plancherel formula for p-adic groups to show that the coefficient of $T_x$ in $t_w$ is a rational function of $\mathbf {q}$, with denominator depending only on the two-sided cell containing w, and dividing a power of the Poincaré polynomial of the finite Weyl group. As an application, we conjecture that these denominators encode more detailed information about the failure of the Kazhdan-Lusztig classification at roots of the Poincaré polynomial than is currently known.

Along the way, we show that upon specializing $\mathbf {q}=q>1$, the map from J to the Harish-Chandra Schwartz algebra is injective. As an application of injectivity, we give a novel criterion for an Iwahori-spherical representation to have fixed vectors under a larger parahoric subgroup in terms of its Kazhdan-Lusztig parameter.

Given a finite abelian group $G$ and $t\in \mathbb{N}$, there are two natural types of subsets of the Cartesian power $G^t$; namely, Cartesian powers $S^t$ where $S$ is a subset of $G$ and (cosets of) subgroups $H$ of $G^t$. A basic question is whether two such sets intersect. In this paper, we show that this decision problem is NP-complete. Furthermore, for fixed $G$ and $S$, we give a complete classification: we determine conditions for when the problem is NP-complete and show that in all other cases the problem is solvable in polynomial time. These theorems play a key role in the classification of algebraic decision problems in finitely generated rings developed in later work of the author.

For Γ a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$ and $n \geq 1$, we study the fibres of the Procesi bundle over the Γ-fixed points of the Hilbert scheme of n points in the plane. For each irreducible component of this fixed point locus, our approach reduces the study of the fibres of the Procesi bundle, as an $(\mathfrak{S}_n \times \Gamma)$-module, to the study of the fibres of the Procesi bundle over an irreducible component of dimension zero in a smaller Hilbert scheme. When Γ is of type A, our main result shows, as a corollary, that the fibre of the Procesi bundle over the monomial ideal associated with a partition λ is induced, as an $(\mathfrak{S}_n \times \Gamma)$-module, from the fibre of the Procesi bundle over the monomial ideal associated with the core of λ. We give different proofs of this corollary in two edge cases using only representation theory and symmetric functions.

We give a new criterion which guarantees that a free group admits a bi-ordering that is invariant under a given automorphism. As an application, we show that the fundamental group of the “magic manifold” is bi-orderable, answering a question of Kin and Rolfsen.

We gather evidence on a new local-global conjecture of Moretó and Rizo on values of irreducible characters of finite groups. For this we study subnormalisers and picky elements in finite groups of Lie type and determine them in many cases, for unipotent elements as well as for semisimple elements of prime power order. We also discuss subnormalisers of unipotent and semisimple elements in connected as well as in disconnected reductive linear algebraic groups.

The growth of central polynomials for matrix algebras over a field of characteristic zero was first studied by Regev in $2016$. This problem can be generalized by analyzing the behavior of the dimension $c_n^z(A)$ of the space of multilinear polynomials of degree n modulo the central polynomials of an algebra A. In $2018$, Giambruno and Zaicev established the existence of the limit $\lim \limits _{n \to \infty }\sqrt [n]{c_n^{z}(A)}.$ In this article, we extend this framework to superalgebras equipped with a superinvolution, proving both the existence and the finiteness of the corresponding limit.

We compare the marked length spectra of some pairs of proper and cocompact cubical actions of a nonvirtually cyclic group on $\mathrm {CAT}(0)$ cube complexes. The cubulations are required to be virtually co-special, have the same sets of convex-cocompact subgroups, and admit a contracting element. There are many groups for which these conditions are always fulfilled for any pair of cubulations, including nonelementary cubulable hyperbolic groups, many cubulable relatively hyperbolic groups, and many right-angled Artin and Coxeter groups.

For these pairs of cubulations, we study the Manhattan curve associated to their combinatorial metrics. We prove that this curve is analytic and convex, and a straight line if and only if the marked length spectra are homothetic. The same result holds if we consider invariant combinatorial metrics in which the lengths of the edges are not necessarily one. In addition, for their standard combinatorial metrics, we prove a large deviations theorem with shrinking intervals for their marked length spectra. We deduce the same result for pairs of word metrics on hyperbolic groups.

The main tool is the construction of a finite-state automaton that simultaneously encodes the marked length spectra of both cubulations in a coherent way, in analogy with results about (bi)combable functions on hyperbolic groups by Calegari-Fujiwara [14]. The existence of this automaton allows us to apply the machinery of thermodynamic formalism for suspension flows over subshifts of finite type, from which we deduce our results.

We first extend previous results of Koskivirta with Wedhorn and Goldring regarding the existence of $\mu $-ordinary Hasse invariants for Hodge-type Shimura varieties to other automorphic line bundles. We also determine exactly which line bundles admit nonzero sections on the stack of G-zips of Pink–Wedhorn–Ziegler. Then, we define and study the Cox ring of the stack of G-zips and show that it is always finitely generated. Finally, beyond the case of line bundles, we define a ring of vector-valued automorphic forms on the stack of G-zips and study its properties. We prove that it is finitely generated in certain cases.

Let $G = K \rtimes \langle t \rangle $ be a finitely generated group where K is abelian and $\langle t\rangle$ is the infinite cyclic group. Let R be a finite symmetric subset of K such that $S = \{ (r,1),(0,t^{\pm 1}) \mid r \in R \}$ is a generating set of G. We prove that the spherical conjugacy ratio, and hence the conjugacy ratio, of G with respect to S is 0 unless G is virtually abelian, confirming a conjecture of Ciobanu, Cox and Martino in this case. We also show that the Baumslag–Solitar group $\mathrm{BS}(1,k)$, $k\geq 2$, has a one-sided Følner sequence F such that the conjugacy ratio with respect to F is non-zero, even though $\mathrm{BS}(1,k)$ is not virtually abelian. This is in contrast to two-sided Følner sequences, where Tointon showed that the conjugacy ratio with respect to a two-sided Følner sequence is positive if and only if the group is virtually abelian.

We say that a semigroup of matrices has a submultiplicative spectrum if the spectrum of the product of any two elements of the semigroup is contained in the product of the two spectra in question (as sets). In this note, we explore an approximate version of this condition.

Jespers and Sun conjectured in [27] that if a finite group G has the property ND, i.e. for every nilpotent element n in the integral group ring $\mathbb{Z}G$ and every primitive central idempotent $e \in \mathbb{Q}G$ one still has $ne \in \mathbb{Z}G$, then at most one of the simple components of the group algebra $\mathbb{Q} G$ has reduced degree bigger than 1. With the exception of one very special series of groups we are able to answer their conjecture, showing that it is true—up to exactly one exception. To do so, we first classify groups with the so-called SN property which was introduced by Liu and Passman in their investigation of the Multiplicative Jordan Decomposition for integral group rings.

The conjecture of Jespers and Sun can also be formulated in terms of a group q(G) made from the group generated by the unipotent units, which is trivial if and only if the ND property holds for the group ring. We answer two more open questions about q(G) and notice that this notion allows to interpret the studied properties in the general context of linear semisimple algebraic groups. Here we show that q(G) is finite for lattices of big rank but can contain elements of infinite order in small rank cases.

We then study further two properties which appeared naturally in these investigations. A first which shows that property ND has a representation theoretical interpretation, while the other can be regarded as indicating that it might be hard to decide ND. Among others we show these two notions are equivalent for groups with SN.

We prove that the Drinfeld center $\mathcal {Z}(\operatorname {Vec}^{\omega }_{A_5})$ of the pointed category associated with the alternating group $A_5$ is the unique example of a perfect weakly group-theoretical modular category of Frobenius–Perron dimension less than $14400$.