We construct efficient topological cobordisms between torus links and large connected sums of trefoil knots. As an application, we show that the signature invariant $\sigma_\omega$ at $\omega=\zeta_6$ takes essentially minimal values on torus links among all concordance homomorphisms with the same normalisation on the trefoil knot.

We provide new upper bounds for sums of certain arithmetic functions in many variables at polynomial arguments and, exploiting recent progress on the mean-value of the Erdős—Hooley $\Delta$-function, we derive lower bounds for the cardinality of those integers not exceeding a given limit that are expressible as certain sums of powers.

Given a morphism $\varphi \;:\; G \to A \wr B$ from a finitely presented group G to a wreath product $A \wr B$, we show that, if the image of $\varphi$ is a sufficiently large subgroup, then $\mathrm{ker}(\varphi)$ contains a non-abelian free subgroup and $\varphi$ factors through an acylindrically hyperbolic quotient of G. As direct applications, we classify the finitely presented subgroups in $A \wr B$ up to isomorphism and we deduce that a finitely presented group having a wreath product $(\text{non-trivial}) \wr (\text{infinite})$ as a quotient must be SQ-universal (extending theorems of Baumslag and Cornulier–Kar). Finally, we exploit our theorem in order to describe the structure of the automorphism groups of several families of wreath products, highlighting an interesting connection with the Kaplansky conjecture on units in group rings.

We study the freeness problem for multiplicative subgroups of $\operatorname{SL}_2(\mathbb{Q})$. For $q = r/p$ in $\mathbb{Q} \cap (0,4)$, where p is prime and $\gcd(r,p)=1$, we initiate the study of the algebraic structure of the group $\Delta_q$ generated by

\[A = \begin{pmatrix}1 & 0 \\ 1 & 1\end{pmatrix} \text{ and } Q_q = \begin{pmatrix}1 & q \\ 0 & 1\end{pmatrix}.\]

$\Delta_{r/p} = \overline{\Gamma}_1^{(p)}(r)$

$\operatorname{SL}_2(\mathbb{Z}[{1}/{p}])$

$r \leq 4$

$r = 5$

$r \in \{ p-1, p+1, (p+1)/2 \}$

$\Delta_{r/p}$

$J_2(r)$

$\operatorname{SL}_2(\mathbb{Z}[{1}/{p}])$

$J_2(r)$

Gay and Meier asked if a trisection diagram for the Gluck twist on a spun or twist-spun 2-knot in $S^4$ obtained by a certain method is standard. In this paper, we show that the trisection diagram for the Gluck twist on the spun $(p+1,p)$-torus knot is standard, where p is any integer greater than or equal to 2.

Let G be a locally compact, Hausdorff, second countable groupoid and A be a separable, $C_0(G^{(0)})$-nuclear, G-$C^*$-algebra. We prove the existence of quasi-invariant, completely positive and contractive lifts for equivariant, completely positive and contractive maps from A into a separable, quotient $C^*$-algebra. Along the way, we construct the Busby invariant for G-actions.

We consider the associated graded $\bigoplus_{k\geq 1} \Gamma_k \mathcal{I} /\Gamma_{k+1} \mathcal{I} $ of the lower central series $\mathcal{I}\,=\,\Gamma_1 \mathcal{I}\supset \Gamma_2 \mathcal{I}\supset \Gamma_3 \mathcal{I} \supset \cdots$ of the Torelli group $\mathcal{I}$ of a compact oriented surface. Its degree-one part is well understood by D. Johnson’s seminal works on the abelianization of the Torelli group. The knowledge of the degree-two part $(\Gamma_2 \mathcal{I} / \Gamma_3 \mathcal{I})\otimes \mathbb{Q}$ with rational coefficients arises from works of S. Morita on the Casson invariant and R. Hain on the Malcev completion of $\mathcal{I}$. Here, we prove that the abelian group $\Gamma_2 \mathcal{I} / \Gamma_3 \mathcal{I}$ is torsion-free, and we describe it as a lattice in a rational vector space. As an application, the group $\mathcal{I}/\Gamma_3 \mathcal{I}$ is computed, and it is shown to embed in the group of homology cylinders modulo the surgery relation of $Y_3$-equivalence.

We extend the work of N. Zubrilina on murmuration of modular forms to the case when prime-indexed coefficients are replaced by squares of primes. Our key observation is that the shape of the murmuration density is the same.

For each closed subtorus T of $(\mathbb{R}/\mathbb{Z})^n$, let D(T) denote the (infimal) $L^\infty$-distance from T to the point $(1/2,\ldots, 1/2)$. The nth Lonely Runner spectrum $\mathcal{S}(n)$ is defined to be the set of all values achieved by D(T) as T ranges over the 1-dimensional subtori of $(\mathbb{R}/\mathbb{Z})^n$ that are not contained in the coordinate hyperplanes. The Lonely Runner Conjecture predicts that $\mathcal{S}(n) \subseteq [0,1/2-1/(n+1)]$. Rather than attack this conjecture directly, we study the qualitative structure of the sets $\mathcal{S}(n)$ via their accumulation points. This project brings into the picture the analogues of $\mathcal{S}(n)$ where 1-dimensional subtori are replaced by k-dimensional subtori or k-dimensional subgroups.

A pattern knot in a solid torus defines a self-map of the smooth knot concordance group. We prove that if the winding number of a pattern is even but not divisible by 8, then the corresponding map is not a homomorphism, thus partially establishing a conjecture of Hedden.

We study the transfer of (co)silting objects in derived categories of module categories via the extension functors induced by a morphism of commutative rings. It is proved that the extension functors preserve (co)silting objects of (co)finite type. In many cases the bounded silting property descends along faithfully flat ring extensions. In particular, the notion of bounded silting complex is Zariski local.

We provide upper bounds for the Assouad spectrum $\dim_A^\theta(\mathrm{Gr}({\kern2pt}f))$ of the graph of a real-valued Hölder or Sobolev function f defined on an interval $I \subset \mathbb{R}$. We demonstrate via examples that all of our bounds are sharp. In the setting of Hölder graphs, we further provide a geometric algorithm which takes as input the graph of an $\alpha$-Hölder continuous function satisfying a matching lower oscillation condition with exponent $\alpha$ and returns the graph of a new $\alpha$-Hölder continuous function for which the Assouad $\theta$-spectrum realizes the stated upper bound for all $\theta\in (0,1)$. Examples of functions to which this algorithm applies include the continuous nowhere differentiable functions of Weierstrass and Takagi.

We show how finiteness properties of a group and a subgroup transfer to finiteness properties of the Schlichting completion relative to this subgroup.n Further, we provide a criterion when the dense embedding of a discrete group into the Schlichting completion relative to one of its subgroups induces an isomorphism in (continuous) cohomology. As an application, we show that the continuous cohomology of the Neretin group vanishes in all positive degrees.

Given r non-zero rational numbers $a_1, \ldots, a_r$ which are not $\pm1$, we complete, under Hypothesis H, a characterisation of the Schinzel–Wójcik r-rational tuples (i.e. r-tuples of rational numbers for which the Schinzel–Wójcik problem has an affirmative answer) which satisfy that the sum of the exponents of the positive elements $a_i$ in the representation of $-1$ in terms of the elements $a_i$ in the multiplicative group $\langle a_1,\dots, a_r\rangle\subset \mathbb{Q}^*$ is even whenever $-1 \in \langle a_1,\dots, a_r\rangle.$

We prove that the initial degenerations of the flag variety admit closed immersions into finite inverse limits of flag matroid strata, where the diagrams are derived from matroidal subdivisions of a suitable flag matroid polytope. As an application, we prove that the initial degenerations of $\mathrm{F}\ell^{\circ}(n)$–the open subvariety of the complete flag variety $\mathrm{F}\ell(n)$ consisting of flags in general position—are smooth and irreducible when $n\leq 4$. We also study the Chow quotient of $\mathrm{F}\ell(n)$ by the diagonal torus of $\textrm{PGL}(n)$ and show that, for $n=4$, this is a log crepant resolution of its log canonical model.

We consider the self-similar measure $\nu_\lambda=\text{law}\left(\sum_{j \geq 0} \xi_j \lambda^j\right)$ on $\mathbb{R}$, where $|\lambda| \lt 1$ and the $\xi_j \sim \nu$ are independent, identically distributed with respect to a measure $\nu$ finitely supported on $\mathbb{Z}$. One example of such a measure is a Bernoulli convolution. It is known that for certain combinations of algebraic $\lambda$ and $\nu$ uniform on an interval, $\nu_\lambda$ is absolutely continuous and its Fourier transform has power decay; in the proof, it is exploited that for these combinations, a quantity called the Garsia entropy $h_{\lambda}(\nu)$ is maximal.

In this paper, we show that the phenomenon of $h_{\lambda}(\nu)$ being maximal is equivalent to absolute continuity of a self-affine measure $\mu_\lambda$, which is naturally associated to $\lambda$ and projects onto $\nu_\lambda$. We also classify all combinations for which this phenomenon occurs: we find that if an algebraic $\lambda$ without a Galois conjugate of modulus exactly one has a $\nu$ such that $h_{\lambda}(\nu)$ is maximal, then all Galois conjugates of $\lambda$ must be smaller in modulus than one and $\nu$ must satisfy a certain finite set of linear equations in terms of $\lambda$. Lastly, we show that in this case, the measure $\mu_\lambda$ is not only absolutely continuous but also has power Fourier decay, which implies the same for $\nu_\lambda$.

We investigate the possibility of defining meaningful upper and lower quantization dimensions for a compactly supported Borel probability measure of order r, including negative values of r. To this end, we employ the concept of partition functions, which generalises the notion of the $L^q$-spectrum, thus extending the authors’ earlier work with Sanguo Zhu in a natural way. In particular, we derive inherent fractal-geometric bounds and easily verifiable necessary conditions for the existence of quantization dimensions. We state the exact asymptotics of the quantization error of negative order for absolutely continuous measures, thereby providing an affirmative answer to an open question regarding the geometric mean error posed by Graf and Luschgy in this journal in 2004.

We prove that there is an absolute constant $C{\,\gt\,}0$ such that every k-vertex connected rainbow graph R with minimum degree at least $C\log k$ has inducibility $k!/(k^k-k)$. The same result holds if $k\ge 11$, and R is a clique. This answers a question posed by Huang, that is a generalisation of an old problem of Erdös and Sós. It remains open to determine the minimum k for which this is true.

Let $X_H$ be the number of copies of a fixed graph H in G(n,p). In 2016, Gilmer and Kopparty conjectured that a local central limit theorem should hold for $X_H$ as long as H is connected, $p\gg n^{-1/m(H)}$ and $n^2(1-p)\gg 1$, where m(H) denotes the m-density of H. Recently, Sah and Sawhney showed that the Gilmer–Kopparty conjecture holds for constant p. In this paper, we show that the Gilmer–Kopparty conjecture holds for triangle counts in the sparse range. More precisely, if $p \in (4n^{-1/2}, 1/2)$, then

\[ {} {} {}\sup_{x\in \mathcal{L}}\left| \dfrac{1}{\sqrt{2\pi}}e^{-x^2/2}-\sigma\cdot \mathbb{P}(X^* = x)\right|=n^{-1/2+o(1)}p^{1/2}, {} {}\]

$\sigma^2 = \mathbb{V}\text{ar}(X_{K_3})$

$X^{*}=(X_{K_3}-\mathbb{E}(X_{K_3}))/\sigma$

$\mathcal{L}$

$X^*$

$n^{-1}\ll p \lt c$

$c\in (0,1)$

$\ell_1$

$m_2$

We give a new criterion which guarantees that a free group admits a bi-ordering that is invariant under a given automorphism. As an application, we show that the fundamental group of the “magic manifold” is bi-orderable, answering a question of Kin and Rolfsen.