We consider the boundary dynamics of iterated function systems of holomorphic self-maps of the unit disc. Our main result provides a sufficient condition which guarantees that the dynamical behaviour of a left iterated function system in the interior of the unit disc can be extended to the boundary. This generalises an extension of the classical Denjoy–Wolff theorem, due to Bourdon, Matache and Shapiro, to the setting of iterated function systems. To do so, we modify estimates for the Hardy norm of composition operators and combine them with a new technique of perturbing a left iterated function system by elliptic Möbius transformations.

We prove that if $p \geq n^{-(q-6)/2}$, then asymptotically almost surely the binomial random q-uniform hypergraph $\mathcal{G}^{(q)}(n,p)$ contains an (n, q, 2)-Steiner system, provided n satisfies the necessary divisibility conditions.

Given a countable group A with trivial bounded cohomology for all semi-separable coefficients and any group $\Gamma$, we prove that the bounded cohomology of certain restricted verbal wreath products $\Gamma \wr^{_W} A$ vanishes in positive degrees, for all semi-separable coefficients. Examples of such verbal wreath products include the standard restricted wreath product (extending a recent result by Monod for lamplighters groups) as well as verbal wreath products arising from n-solvable, n-nilpotent and k-Burnside ($k = 2,3,4,6$) verbal products. Moreover, we prove that the stable commutator length of any verbal wreath product $\Gamma \wr^{_W} A$ vanishes. As an application, we show that every group of type $F_p$ isometrically embeds into a group of type $F_p$ with vanishing bounded cohomology in degree $n\geq 1$ for all semi-separable coefficients.

Let v(k) be the smallest integer larger than 1 that does not occur among the denominators in any identity of the form

\begin{align*}1=\frac1{n_1}+\cdots+\frac1{n_k},\end{align*}

$1 \leq n_1\lt\cdots\lt n_k$

$v(k)\gg k!$

$c\gt0$

\begin{align*}v(k)\ge e^{c k^2}\end{align*}

In the homology cobordism group $\Theta_\mathbb{Z}^3$, it is not known if there are non-trivial linear dependences between Seifert fibered spheres. Based on involutive Heegaard Floer theory, Hendricks, Manolescu and Zemke introduced the local equivalence group $\mathfrak{I}$ along with the homomorphism $h\;:\;\Theta_\mathbb{Z}^3 \rightarrow \mathfrak{I}$. Using the work of Dai and Stoffregen, one can find non-trivial linear dependences between the images of Seifert fibered spheres under h. Therefore, it is interesting to ask if such dependences in $\mathfrak{I}$ originate from $\Theta_\mathbb{Z}^3$. In this paper, by employing the $r_s$-invariants from the filtered instanton Floer homology developed by Nozaki, Sato and Taniguchi, we provide certain conditions to guarantee that such relations are not realised even in the rational homology cobordism group. We also discuss the local equivalence class of the $\operatorname{Pin(2)}$-equivariant Seiberg–Witten Floer stable homotopy type.

We revisit the theory of Ihara L-functions in the context initially studied by Bass and Hashimoto and more recently by Zakharov. In particular, we study whether the Artin formalism is satisfied by these L-functions. As an application, we give a proof of the analogue of Iwasawa’s asymptotic class number formula for the p-part of the number of spanning trees in branched $\mathbb{Z}_{p}$-towers of finite connected graphs using Ihara zeta and L-functions. Moreover, we relate a generator for the characteristic ideal of the finitely generated torsion Iwasawa module that governs the growth of the p-part of the number of spanning trees in such towers with Ihara L-functions.

We study the geometry of the complex hyperbolic triangle group $\Delta_{4,\infty,\infty;\infty}$ acting on the complex hyperbolic plane $\mathbf{H}^2_{\mathbb C}$ and show that the 3-manifold at infinity of $\Delta_{4,\infty,\infty;\infty}$ is the complement of the chain link $8^4_1$ in the 3-sphere. In particular, the four-cusped hyperbolic 3-manifold $S^3-8^4_1$ admits a spherical CR-uniformisation. Since the ideal boundary of the Ford domain is an infinite-genus handlebody, we introduce a new technique to construct embedded topological disks that cap off the genus, ensuring their boundaries are compatible with the side-pairings. This surgical approach facilitates the determination of the fundamental group of the manifold at infinity and simplifies the computation compared to [14].

Based on this result and previous studies, we conjecture that the chain link $C_p$ serves as an ancestor of the 3-manifold at infinity for the critical complex hyperbolic triangle group $\Delta_{p,q,r;\infty}$, where $3 \leq p \leq 9$.

Given any link $L\subseteq S^3$, we show that it is possible to embed an unknot U in its complement so that the link $L\cup U$ satisfies the Meridional Rank Conjecture (MRC). The bridge numbers in our construction fit into the equality $\beta(L\cup U)=2\beta(L)-1=\text{rank}(\pi_1(S^3\backslash (L\cup U)))$. In addition, we prove the MRC for new infinite families of links and distinguish them from previously settled cases through an application of bridge distance.

In this paper, we formulate a contact analogue on the one-jet bundle $J^1B$ of Weinstein’s observation which says the classical action functional on the cotangent bundle is a generating function of any Hamiltonian isotope of the zero section. We do this by identifying the correct action functional which is defined on the space of Hamiltonian-translated (piecewise smooth) horizontal curves of the contact distribution, which we call the Carnot path space. Then we give a canonical construction of generating functions of Legendrians in the 1-jet bundle (contact isotopic to the zero section), which is the Legendrian counterpart of Laudenbach–Sikorav’s canonical construction of the generating function of Hamiltonian isotope of the zero section on the cotangent bundle which utilises a finite dimensional approximation of the action functional.

Motivated by this construction, we develop a Floer theoretic construction of spectral invariants for the Legendrian submanifolds in sequel which is the contact analogue to the construction given in [33, 34] for the Lagrangian submanifolds in the cotangent bundle.

We prove that, if G is a finite almost simple group and H is a maximal subgroup of G, then the 10th term of the derived series of H is perfect. The same is true if G is perfect and H is core-free. The constant 10 is best possible.