We consider a family of cyclic presentations and show that, subject to certain conditions on the defining parameters, they are spines of closed 3-manifolds. These are new examples where the reduced Whitehead graphs are of the same type as those of the Fractional Fibonacci presentations; here the corresponding manifolds are often (but not always) hyperbolic. We also express a lens space construction in terms of a class of positive cyclic presentations that are spines of closed 3-manifolds. These presentations then furnish examples where the Whitehead graphs are of the same type as those of the positive cyclic presentations of type $\mathfrak {Z}$, as considered by McDermott.

We show that the group $ \langle a,b,c,t \,:\, a^t=b,b^t=c,c^t=ca^{-1} \rangle$ is profinitely rigid amongst free-by-cyclic groups, providing the first example of a hyperbolic free-by-cyclic group with this property.

Using a recent result of Bowden, Hensel and Webb, we prove the existence of a homeomorphism with positive stable commutator length in the group of homeomorphisms of the Klein bottle which are isotopic to the identity.

A classical result of Reinhold Baer states that a group G = XN, which is the product of two normal supersoluble subgroups X and N, is supersoluble if and only if Gʹ is nilpotent. This result has been weakened in [6] for a finite group G: in fact, we do not need that both X and N are normal, but only that N is normal and X permutes with every maximal subgroup of each Sylow subgroup of N.

In our paper, we improve the result mentioned above by showing that we only need X to permute with the maximal subgroups of the non-cyclic Sylow subgroups of N. Also, we extend this result (and several others) to relevant classes of infinite groups.

The central idea behind our results stems from grasping the key aspects of what happens in [6]. It turns out that tensor products play a very crucial role, and it is precisely this shift of perspective that makes it possible not only to improve those results but also extend them to infinite groups.

Given a word $w(x_{1},\ldots,x_{r})$, i.e. an element in the free group on r elements, and an integer $d\geq1$, we study the characteristic polynomial of the random matrix $w(X_{1},\ldots,X_{r})$, where $X_{i}$ are Haar-random independent $d\times d$ unitary matrices. If $c_{m}(X)$ denotes the mth coefficient of the characteristic polynomial of X, our main theorem implies that there is a positive constant $\epsilon(w)$, depending only on w, such that

\[|\mathbb{E}(c_{m}(w(X_{1},\ldots,X_{r})))|\leq\binom{d}{m}^{\!\!1-\epsilon(w)},\]

$1\leq m\leq d$

$\mathbb{E}(c_{m}(w))$

We find an Lascoux–Leclerc–Thibon (LLT)-type formula for a general power of the nabla operator of [BG99] applied to the Cauchy product for the modified Macdonald polynomials, and use it to deduce a new proof of the generalized shuffle theorem describing $\nabla^k e_n$ [HHL+05a, CM18, Mel21], and the formula for $(\nabla^k p_1^n,e_n)$ from [EH16, GH22] as corollaries. We give a direct proof of the theorem by verifying that the LLT expansion satisfies the defining properties of $\nabla^k$, such as triangularity in the dominance order, as well as a geometric proof based on a method for counting bundles on $\mathbb{P}^1$ due to the second author [Mel20]. These formulas are related to an affine paving of the type A unramified affine Springer fiber studied by Goresky, Kottwitz, and MacPherson in [GKM04], and also to Stanley’s chromatic symmetric functions.

We prove several results showing that every locally finite Borel graph whose large-scale geometry is ‘tree-like’ induces a treeable equivalence relation. In particular, our hypotheses hold if each component of the original graph either has bounded tree-width or is quasi-isometric to a tree, answering a question of Tucker-Drob. In the latter case, we moreover show that there exists a Borel quasi-isometry to a Borel forest, under the additional assumption of (componentwise) bounded degree. We also extend these results on quasi-treeings to Borel proper metric spaces. In fact, our most general result shows treeability of countable Borel equivalence relations equipped with an abstract wallspace structure on each class obeying some local finiteness conditions, which we call a proper walling. The proof is based on the Stone duality between proper wallings and median graphs (i.e., CAT(0) cube complexes). Finally, we strengthen the conclusion of treeability in these results to hyperfiniteness in the case where the original graph has one (selected) end per component, generalizing the same result for trees due to Dougherty–Jackson–Kechris.

Let W be a group endowed with a finite set S of generators. A representation $(V,\rho )$ of W is called a reflection representation of $(W,S)$ if $\rho (s)$ is a (generalized) reflection on V for each generator $s \in S$. In this article, we prove that for any irreducible reflection representation V, all the exterior powers $\bigwedge ^d V$, $d = 0, 1, \dots , \dim V$, are irreducible W-modules, and they are non-isomorphic to each other. This extends a theorem of R. Steinberg which is stated for Euclidean reflection groups. Moreover, we prove that the exterior powers (except for the 0th and the highest power) of two non-isomorphic reflection representations always give non-isomorphic W-modules. This allows us to construct numerous pairwise non-isomorphic irreducible representations for such groups, especially for Coxeter groups.

In a paper from 1980, Shelah constructed an uncountable group all of whose proper subgroups are countable. Assuming the continuum hypothesis, he constructed an uncountable group G that moreover admits an integer n satisfying that for every uncountable $X\subseteq G$, every element of G may be written as a group word of length n in the elements of X. The former is called a Jónsson group, and the latter is called a Shelah group.

In this paper, we construct a Shelah group on the grounds of $\textsf {{ZFC}}$ alone – that is, without assuming the continuum hypothesis. More generally, we identify a combinatorial condition (coming from the theories of negative square-bracket partition relations and strongly unbounded subadditive maps) sufficient for the construction of a Shelah group of size $\kappa $, and we prove that the condition holds true for all successors of regular cardinals (such as $\kappa =\aleph _1,\aleph _2,\aleph _3,\ldots $). This also yields the first consistent example of a Shelah group of size a limit cardinal.

We prove that any compact R-analytic group is linear when R is a pro-p domain of characteristic zero.

We give a complete description of Rees quotients of free inverse semigroups given by positive relators that satisfy nontrivial identities, including identities in signature with involution. They are finitely presented in the class of all inverse semigroups. Those that satisfy a nontrivial semigroup identity have polynomial growth and can be given by an irredundant presentation with at most four relators. Those that satisfy a nontrivial identity in signature with involution, but which do not satisfy a nontrivial semigroup identity, have exponential growth and fall within two infinite families of finite presentations with two generators. The first family involves an unbounded number of relators and the other involves presentations with at most four relators of unbounded length. We give a new sufficient condition for which a finite set X of reduced words over an alphabet $A\cup A^{-1}$ freely generates a free inverse subsemigroup of $FI_A$ and use it in our proofs.

Given a presentation of a monoid $M$, combined work of Pride and of Guba and Sapir provides an exact sequence connecting the relation bimodule of the presentation (in the sense of Ivanov) with the first homology of the Squier complex of the presentation, which is naturally a $\mathbb ZM$-bimodule. This exact sequence was used by Kobayashi and Otto to prove the equivalence of Pride’s finite homological type property with the homological finiteness condition bi-$\mathrm {FP}_3$. Guba and Sapir used this exact sequence to describe the abelianization of a diagram group. We prove here a generalization of this exact sequence of bimodules for presentations of associative algebras. Our proof is more elementary than the original proof for the special case of monoids.

We study linear random walks on the torus and show a quantitative equidistribution statement, under the assumption that the Zariski closure of the acting group is semisimple.

In an earlier work, we defined a “generalised Temperley–Lieb algebra” $TL_{r, 1, n}$ corresponding to the imprimitive reflection group G(r, 1, n) as a quotient of the cyclotomic Hecke algebra. In this work we introduce the generalised Temperley–Lieb algebra $TL_{r, p, n}$ which corresponds to the complex reflection group G(r, p, n). Our definition identifies $TL_{r, p, n}$ as the fixed-point subalgebra of $TL_{r, 1, n}$ under a certain automorphism $\sigma$. We prove the cellularity of $TL_{r, p, n}$ by proving that $\sigma$ induces a special shift automorphism with respect to the cellular structure of $TL_{r, 1, n}$. We also give a description of the cell modules of $TL_{r, p, n}$ and their decomposition numbers, and finally we point to how our algebras might be categorified and could lead to a diagrammatic theory.

The space of monic squarefree complex polynomials has a stratification according to the multiplicities of the critical points. We introduce a method to study these strata by way of the infinite-area translation surface associated to the logarithmic derivative $df/f$ of the polynomial. We determine the monodromy of these strata in the braid group, thus describing which braidings of the roots are possible if the orders of the critical points are required to stay fixed. Mirroring the story for holomorphic differentials on higher-genus surfaces, we find the answer is governed by the framing of the punctured disk induced by the horizontal foliation on the translation surface.

We provide two constructions of hyperbolic metrics on 3-manifolds with Heegaard splittings that satisfy certain topological conditions, which both apply to random Heegaard splitting with asymptotic probability 1. These constructions provide a lot of control on the resulting metric, allowing us to prove various results about the coarse growth rate of geometric invariants, such as diameter and injectivity radius, and about arithmeticity and commensurability in families of random 3-manifolds. For example, we show that the diameter of a random Heegaard splitting grows coarsely linearly in the length of the associated random walk. The constructions only use tools from the deformation theory of Kleinian groups, that is, we do not rely on the solution of the geometrization conjecture by Perelman. In particular, we give a proof of Maher’s result that random 3-manifolds are hyperbolic that bypasses geometrization.

We prove that a homomorphism between free groups of finite rank equipped with the bi-invariant word metrics associated with finite generating sets is a quasi-isometry if and only if it is an isomorphism.

The minimal faithful permutation degree $\mu (G)$ of a finite group G is the least integer n such that G is isomorphic to a subgroup of the symmetric group $S_n$. If G has a normal subgroup N such that $\mu (G/N)> \mu (G)$, then G is exceptional. We prove that the proportion of exceptional groups of order $p^6$ for primes $p \geq 5$ is asymptotically zero. We identify $(11p+107)/2$ such groups and conjecture that there are no others.

This paper studies reversibility and transitivity of semigroups acting on homogeneous spaces. Properties of the reversor set of a subsemigroup acting on homogeneous spaces are presented. Let G be a topological group and L a subgroup of G. Assume that S is a subsemigroup of G with nonempty interior. It is presented a study of the reversibility of the S-action on $G/L$ in terms of the actions of S and L on homogeneous spaces of G. The results relate the reversibility and the transitivity of S in $G/L$ with the minimality of the action of L on homogeneous spaces of G. In addition, sufficient conditions for S to be right reversible in G if S is reversible in $G/L$ are presented.

Inspired by work of Szymik and Wahl on the homology of Higman–Thompson groups, we establish a general connection between ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces, based on the construction of small permutative categories of compact open bisections. This allows us to analyse homological invariants of topological full groups in terms of homology for ample groupoids.

Applications include complete rational computations, general vanishing and acyclicity results for group homology of topological full groups as well as a proof of Matui’s AH-conjecture for all minimal, ample groupoids with comparison.