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.

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

Let e$e$ and q$q$ be fixed co-prime integers satisfying 1\lt e\lt q$1\lt e\lt q$. Let \mathscr {C}$\mathscr {C}$ be a certain family of deformations of the curve y^e=x^q$y^e=x^q$. That family is called the (e,q)$(e,q)$-curve and is one of the types of curves called plane telescopic curves. Let \varDelta$\varDelta$ be the discriminant of \mathscr {C}$\mathscr {C}$. Following pioneering work by Buchstaber and Leykin (BL), we determine the canonical basis \{ L_j \}$\{ L_j \}$ of the space of derivations tangent to the variety \varDelta =0$\varDelta =0$ and describe their specific properties. Such a set \{ L_j \}$\{ L_j \}$ gives rise to a system of linear partial differential equations (heat equations) satisfied by the function \sigma (u)$\sigma (u)$ associated with \mathscr {C}$\mathscr {C}$, 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.

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.

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

The fine curve graph of a surface was introduced by Bowden, Hensel, and Webb as a graph consisting of essential simple closed curves in the surface. Long, Margalit, Pham, Verberne, and Yao proved that the automorphism group of the fine curve graph of a closed orientable surface is isomorphic to the homeomorphism group of the surface. In this paper, based on their argument, we prove that the automorphism group of the fine curve graph of a closed nonorientable surface N$N$ of genus g \geq 4$g \geq 4$ is isomorphic to the homeomorphism group of N$N$.

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

In Caspers et al. (Can. J. Math. 75[6] [2022], 1–18), transference results between multilinear Fourier and Schur multipliers on noncommutative L_p$L_p$-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 consider the space \mathcal{P}_d$\mathcal{P}_d$ of smooth complex projective plane curves of degree d$d$. There is the tautological family of plane curves defined over \mathcal{P}_d$\mathcal{P}_d$, which has an associated monodromy representation \rho _d: \pi _1(\mathcal{P}_d) \to \textrm{Mod}(\Sigma _g)$\rho _d: \pi _1(\mathcal{P}_d) \to \textrm{Mod}(\Sigma _g)$ into the mapping class group of the fiber. For d \le 4$d \le 4$, classical algebraic geometry implies the surjectivity of \rho _d$\rho _d$. For d \ge 5$d \ge 5$, the existence of a (d-3)^{rd}$(d-3)^{rd}$ root of the canonical bundle implies that \rho _d$\rho _d$ cannot be surjective. The main result of this paper is that for d = 5$d = 5$, the image of \rho _5$\rho _5$ 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)$\textrm{Mod}(\Sigma _g)$.

The monogenic free inverse semigroup FI_1$FI_1$ 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$FI_1$ 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$FI_1$ is finitely presented as a semigroup if and only if it is a finite semilattice.

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

Let M$M$ be an oriented smooth manifold and \operatorname{Homeo}\!(M,\omega )$\operatorname{Homeo}\!(M,\omega )$ the group of measure preserving homeomorphisms of M$M$, where \omega$\omega$ 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 )$\operatorname{Homeo}_0\!(M,\omega )$ and \operatorname{Homeo}_+\!(M,\omega )$\operatorname{Homeo}_+\!(M,\omega )$, 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 ))$\operatorname{H}_b^n(\operatorname{Homeo}_0\!(M,\omega ))$, where M$M$ is a hyperbolic manifold of dimension n$n$.

• • Euler classes in \operatorname{H}_b^2(\operatorname{Homeo}_+(S,\omega ))$\operatorname{H}_b^2(\operatorname{Homeo}_+(S,\omega ))$, where S$S$ 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$3$-manifolds; hence, they are non-trivial.

We show that for n \neq 1,4$n \neq 1,4$, the simplicial volume of an inward tame triangulable open n$n$-manifold M$M$ with amenable fundamental group at infinity at each end is finite; moreover, we show that if also \pi _1(M)$\pi _1(M)$ is amenable, then the simplicial volume of M$M$ vanishes. We show that the same result holds for finitely-many-ended triangulable manifolds which are simply connected at infinity.

The Basilica group is a well-known 2-generated weakly branch, but not branch, group acting on the binary rooted tree. Recently, a more general form of the Basilica group has been investigated by Petschick and Rajeev, which is an s$s$-generated weakly branch, but not branch, group that acts on the m$m$-adic tree, for s,m\ge 2$s,m\ge 2$. A larger family of groups, which contains these generalised Basilica groups, is the family of iterated monodromy groups. With the new developments by Francoeur, the study of the existence of maximal subgroups of infinite index has been extended from branch groups to weakly branch groups. Here we show that a subfamily of iterated monodromy groups, which more closely resemble the generalised Basilica groups, have maximal subgroups only of finite index.

This paper describes how to compute algorithmically certain twisted signature invariants of a knot K$K$ using twisted Blanchfield forms. An illustration of the algorithm is implemented on (2,q)$(2,q)$-torus knots. Additionally, using satellite formulas for these invariants, we also show how to obstruct the sliceness of certain iterated torus knots.

We consider K-theoretic Gromov-Witten theory of root constructions. We calculate some genus 0$0$ K-theoretic Gromov-Witten invariants of a root gerbe. We also obtain a K-theoretic relative/orbifold correspondence in genus 0$0$.

In this article, we study Galois points of plane curves and the extension of the corresponding Galois group to \mathrm{Bir}(\mathbb{P}^2)$\mathrm{Bir}(\mathbb{P}^2)$. We prove that if the Galois group has order at most 3$3$, it always extends to a subgroup of the Jonquières group associated with the point P$P$. Conversely, with a degree of at least 4$4$, we prove that it is false. We provide an example of a Galois extension whose Galois group is extendable to Cremona transformations but not to a group of de Jonquières maps with respect to P$P$. In addition, we also give an example of a Galois extension whose Galois group cannot be extended to Cremona transformations.

Let C\; : \;y^2=f(x)$C\; : \;y^2=f(x)$ be a hyperelliptic curve of genus g\geq 1$g\geq 1$, defined over a complete discretely valued field K$K$, with ring of integers O_K$O_K$. Under certain conditions on C$C$, mild when residue characteristic is not 2$2$, we explicitly construct the minimal regular model with normal crossings \mathcal{C}/O_K$\mathcal{C}/O_K$ of C$C$. In the same setting we determine a basis of integral differentials of C$C$, that is an O_K$O_K$-basis for the global sections of the relative dualising sheaf \omega _{\mathcal{C}/O_K}$\omega _{\mathcal{C}/O_K}$.

Generalizing a result of Wulf-Dieter Geyer in his thesis, we prove that if K$K$ is a finitely generated extension of transcendence degree r$r$ of a global field and A$A$ is a closed abelian subgroup of \textrm{Gal}(K)$\textrm{Gal}(K)$, then {\mathrm{rank}}(A)\le r+1${\mathrm{rank}}(A)\le r+1$. Moreover, if \mathrm{char}(K)=0$\mathrm{char}(K)=0$, then {\hat{\mathbb{Z}}}^{r+1}${\hat{\mathbb{Z}}}^{r+1}$ is isomorphic to a closed subgroup of \textrm{Gal}(K)$\textrm{Gal}(K)$.