We prove the Girth Alternative for finitely generated subgroups of $PL_o(I)$. We also prove that a finitely generated subgroup of Homeo$_{+}(I)$ which is sufficiently rich with hyperbolic-like elements has infinite girth.

Let $\varphi$ be a normal semifinite faithful weight on a von Neumann algebra $A$, let $(\sigma ^\varphi _r)_{r\in {\mathbb R}}$ denote the modular automorphism group of $\varphi$, and let $T\colon A\to A$ be a linear map. We say that $T$ admits an absolute dilation if there exists another von Neumann algebra $M$ equipped with a normal semifinite faithful weight $\psi$, a $w^*$-continuous, unital and weight-preserving $*$-homomorphism $J\colon A\to M$ such that $\sigma ^\psi \circ J=J\circ \sigma ^\varphi$, as well as a weight-preserving $*$-automorphism $U\colon M\to M$ such that $T^k={\mathbb {E}}_JU^kJ$ for all integer $k\geq 0$, where $ {\mathbb {E}}_J\colon M\to A$ is the conditional expectation associated with $J$. Given any locally compact group $G$ and any real valued function $u\in C_b(G)$, we prove that if $u$ induces a unital completely positive Fourier multiplier $M_u\colon VN(G) \to VN(G)$, then $M_u$ admits an absolute dilation. Here, $VN(G)$ is equipped with its Plancherel weight $\varphi _G$. This result had been settled by the first named author in the case when $G$ is unimodular so the salient point in this paper is that $G$ may be nonunimodular, and hence, $\varphi _G$ may not be a trace. The absolute dilation of $M_u$ implies that for any $1\lt p\lt \infty$, the $L^p$-realization of $M_u$ can be dilated into an isometry acting on a noncommutative $L^p$-space. We further prove that if $u$ is valued in $[0,1]$, then the $L^p$-realization of $M_u$ is a Ritt operator with a bounded $H^\infty$-functional calculus.

We study Morita equivalence for idempotent rings with involution. Following the ideas of Rieffel, we define Rieffel contexts, and we also introduce Morita $*$-contexts and enlargements for rings with involution. We prove that two idempotent rings with involution have a joint enlargement if and only if they are connected by a unitary and full Rieffel context. These conditions are also equivalent to having a unitary and surjective Morita $*$-context between those rings. We also examine how the mentioned conditions are connected to the existence of certain equivalence functors between the categories of firm modules over the given rings with involution.

We consider the space $\mathcal{P}_d$ of smooth complex projective plane curves of degree $d$. There is the tautological family of plane curves defined over $\mathcal{P}_d$, which has an associated monodromy representation $\rho _d: \pi _1(\mathcal{P}_d) \to \textrm{Mod}(\Sigma _g)$ into the mapping class group of the fiber. For $d \le 4$, classical algebraic geometry implies the surjectivity of $\rho _d$. For $d \ge 5$, the existence of a $(d-3)^{rd}$ root of the canonical bundle implies that $\rho _d$ cannot be surjective. The main result of this paper is that for $d = 5$, the image of $\rho _5$ is as large as possible, subject to this constraint. This requires combining the algebro-geometric work of Lönne with Johnson’s theory of the Torelli subgroup of $\textrm{Mod}(\Sigma _g)$.

Let $K={\mathbb {Q}}(\sqrt {-7})$ and $\mathcal {O}$ the ring of integers in $K$. The prime $2$ splits in $K$, say $2{\mathcal {O}}={\mathfrak {p}}\cdot {\mathfrak {p}}^*$. Let $A$ be an elliptic curve defined over $K$ with complex multiplication by $\mathcal {O}$. Assume that $A$ has good ordinary reduction at both $\mathfrak {p}$ and ${\mathfrak {p}}^*$. Write $K_\infty$ for the field generated by the $2^\infty$–division points of $A$ over $K$ and let ${\mathcal {G}}={\mathrm {Gal}}(K_\infty /K)$. In this paper, by adopting a congruence formula of Yager and De Shalit, we construct the two-variable $2$-adic $L$-function on $\mathcal {G}$. Then by generalizing De Shalit’s local structure theorem to the two-variable setting, we prove a two-variable elliptic analogue of Iwasawa’s theorem on cyclotomic fields. As an application, we prove that every branch of the two-variable measure has Iwasawa $\mu$ invariant zero.

We study the vectorial length compactification of the space of conjugacy classes of maximal representations of the fundamental group $\Gamma$ of a closed hyperbolic surface $\Sigma$ in $\textrm{PSL}(2,{\mathbb{R}})^n$. We identify the boundary with the sphere ${\mathbb{P}}(({\mathcal{ML}})^n)$, where $\mathcal{ML}$ is the space of measured geodesic laminations on $\Sigma$. In the case $n=2$, we give a geometric interpretation of the boundary as the space of homothety classes of ${\mathbb{R}}^2$-mixed structures on $\Sigma$. We associate to such a structure a dual tree-graded space endowed with an ${\mathbb{R}}_+^2$-valued metric, which we show to be universal with respect to actions on products of two $\mathbb{R}$-trees with the given length spectrum.

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.

Let $M$ be an oriented smooth manifold and $\operatorname{Homeo}\!(M,\omega )$ the group of measure preserving homeomorphisms of $M$, where $\omega$ is a finite measure induced by a volume form. In this paper, we define volume and Euler classes in bounded cohomology of an infinite dimensional transformation group $\operatorname{Homeo}_0\!(M,\omega )$ and $\operatorname{Homeo}_+\!(M,\omega )$, respectively, and in several cases prove their non-triviality. More precisely, we define:

• • Volume classes in $\operatorname{H}_b^n(\operatorname{Homeo}_0\!(M,\omega ))$, where $M$ is a hyperbolic manifold of dimension $n$.

• • Euler classes in $\operatorname{H}_b^2(\operatorname{Homeo}_+(S,\omega ))$, where $S$ is an oriented closed hyperbolic surface.

We show that Euler classes have positive norms for any closed hyperbolic surface and volume classes have positive norms for all hyperbolic surfaces and certain hyperbolic $3$-manifolds; hence, they are non-trivial.

The monogenic free inverse semigroup $FI_1$ is not finitely presented as a semigroup due to the classic result by Schein (1975). We extend this result and prove that a finitely generated subsemigroup of $FI_1$ is finitely presented if and only if it contains only finitely many idempotents. As a consequence, we derive that an inverse subsemigroup of $FI_1$ is finitely presented as a semigroup if and only if it is a finite semilattice.

Let $e$ and $q$ be fixed co-prime integers satisfying $1\lt e\lt q$. Let $\mathscr {C}$ be a certain family of deformations of the curve $y^e=x^q$. That family is called the $(e,q)$-curve and is one of the types of curves called plane telescopic curves. Let $\varDelta$ be the discriminant of $\mathscr {C}$. Following pioneering work by Buchstaber and Leykin (BL), we determine the canonical basis $\{ L_j \}$ of the space of derivations tangent to the variety $\varDelta =0$ and describe their specific properties. Such a set $\{ L_j \}$ gives rise to a system of linear partial differential equations (heat equations) satisfied by the function $\sigma (u)$ associated with $\mathscr {C}$, and eventually gives its explicit power series expansion. This is a natural generalisation of Weierstrass’ result on his sigma function. We attempt to give an accessible description of various aspects of the BL theory. Especially, the text contains detailed proofs for several useful formulae and known facts since we know of no works which include their proofs.

In Caspers et al. (Can. J. Math. 75[6] [2022], 1–18), transference results between multilinear Fourier and Schur multipliers on noncommutative $L_p$-spaces were shown for unimodular groups. We propose a suitable extension of the definition of multilinear Fourier multipliers for non-unimodular groups and show that the aforementioned transference results also hold in this more general setting.

We provide a general recursive method for constructing transfer systems on finite lattices. Using this, we calculate the number of homotopically distinct $N_{\infty} $ operads for dihedral groups $D_{p^n}$, $p \gt 2$ prime, and cyclic groups $C_{qp^n}$, $p \neq q$ prime. We then further display some of the beautiful combinatorics obtained by restricting to certain homotopically meaningful $N_\infty$ operads for these groups.

We show that when a finitely presented Bestvina–Brady group splits as an amalgamated product over a subgroup $H$, its defining graph contains an induced separating subgraph whose associated Bestvina–Brady group is contained in a conjugate of $H$.

Let $F:\; {\mathscr {C}} \to {\mathscr {E}} \ $ be a functor from a category $\mathscr {C} \ $ to a homological (Borceux–Bourn) or semi-abelian (Janelidze–Márki–Tholen) category $\mathscr {E}$. We investigate conditions under which the homology of an object $X$ in $\mathscr {C}$ with coefficients in the functor $F$, defined via projective resolutions in $\mathscr {C}$, remains independent of the chosen resolution. Consequently, the left derived functors of $F$ can be constructed analogously to the classical abelian case.

Our approach extends the concept of chain homotopy to a non-additive setting using the technique of imaginary morphisms. Specifically, we utilize the approximate subtractions of Bourn–Janelidze, originally introduced in the context of subtractive categories. This method is applicable when $\mathscr {C}$ is a pointed regular category with finite coproducts and enough projectives, provided the class of projectives is closed under protosplit subobjects, a new condition introduced in this article and naturally satisfied in the abelian context. We further assume that the functor $F$ meets certain exactness conditions: for instance, it may be protoadditive and preserve proper morphisms and binary coproducts—conditions that amount to additivity when $\mathscr {C}$ and $\mathscr {E}$ are abelian categories.

Within this framework, we develop a basic theory of derived functors, compare it with the simplicial approach, and provide several examples.

A classification of multiplication modules over multiplication rings with finitely many minimal primes is obtained. A characterization of multiplication rings with finitely many minimal primes is given via faithful, Noetherian, distributive modules. It is proven that for a multiplication ring with finitely many minimal primes every faithful, Noetherian, distributive module is a faithful multiplication module, and vice versa.

For finite nilpotent groups $J$ and $N$, suppose $J$ acts on $N$ via automorphisms. We exhibit a decomposition of the first cohomology set in terms of the first cohomologies of the Sylow $p$-subgroups of $J$ that mirrors the primary decomposition of $H^1(J,N)$ for abelian $N$. We then show that if $N \rtimes J$ acts on some non-empty set $\Omega$, where the action of $N$ is transitive and for each prime $p$ a Sylow $p$-subgroup of $J$ fixes an element of $\Omega$, then $J$ fixes an element of $\Omega$.

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.

In this paper, we deal with a weighted eigenvalue problem for the anisotropic $(p,q)$-Laplacian with Dirichlet boundary conditions. We study the main properties of the first eigenvalue and a reverse Hölder type inequality for the corresponding eigenfunctions.

We show that the sets of $d$-dimensional Latin hypercubes over a non-empty set $X$, with $d$ running over the positive integers, determine an operad which is isomorphic to a sub-operad of the endomorphism operad of $X$. We generalise this to categories with finite products, and then further to internal versions for certain Cartesian closed monoidal categories with pullbacks.

Let $(A,\mathfrak{m})$ be a Cohen–Macaulay local ring, and then the notion of a $T$-split sequence was introduced in the part-1 of this paper for the $\mathfrak{m}$-adic filtration with the help of the numerical function $e^T_A$. In this article, we explore the relation between Auslander–Reiten (AR)-sequences and $T$-split sequences. For a Gorenstein ring $(A,\mathfrak{m})$, we define a Hom-finite Krull–Remak–Schmidt category $\mathcal{D}_A$ as a quotient of the stable category $\underline{\mathrm{CM}}(A)$. This category preserves isomorphism, that is, $M\cong N$ in $\mathcal{D}_A$ if and only if $M\cong N$ in $\underline{\mathrm{CM}}(A)$.This article has two objectives: first objective is to extend the notion of $T$-split sequences, and second objective is to explore the function $e^T_A$ and $T$-split sequences. When $(A,\mathfrak{m})$ is an analytically unramified Cohen–Macaulay local ring and $I$ is an $\mathfrak{m}$-primary ideal, then we extend the techniques in part-1 of this paper to the integral closure filtration with respect to $I$ and prove a version of Brauer–Thrall-II for a class of such rings.