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.

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.

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.

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.

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.

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.

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.

We establish new results on complex and $p$-adic linear independence on a class of semiabelian varieties. As applications, we obtain transcendence results concerning complex and $p$-adic Weierstrass sigma functions associated with elliptic curves.

In this paper, we establish a new version of one-dimensional discrete improved Hardy’s inequality with shifts by introducing a shifting discrete Dirichlet’s Laplacian. We prove that the general discrete Hardy’s inequality as well as its variants in some special cases admit improvements. Further, it is proved that two-variable discrete $p$-Hardy inequality can also be improved via improved discrete $p$-Hardy inequality in one dimension. The result is also extended to the multivariable cases.