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Boxes, extended boxes and sets of positive upper density in the Euclidean space Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2021-01-14 POLONA DURCIK; VJEKOSLAV KOVAČ
We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing
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Quasiconformality and hyperbolic skew Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-12-18 COLLEEN ACKERMANN; ALASTAIR FLETCHER
We prove that if $f:\mathbb{B}^n \to \mathbb{B}^n$ , for n ≥ 2, is a homeomorphism with bounded skew over all equilateral hyperbolic triangles, then f is in fact quasiconformal. Conversely, we show that if $f:\mathbb{B}^n \to \mathbb{B}^n$ is quasiconformal then f is η-quasisymmetric in the hyperbolic metric, where η depends only on n and K. We obtain the same result for hyperbolic n-manifolds. Analogous
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Difference sets in higher dimensions Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-12-14 AKSHAT MUDGAL
Let d ≥ 3 be a natural number. We show that for all finite, non-empty sets $A \subseteq \mathbb{R}^d$ that are not contained in a translate of a hyperplane, we have $$\begin{equation*} |A-A| \geq (2d-2)|A| - O_d(|A|^{1- \delta}),\end{equation*}$$ where δ > 0 is an absolute constant only depending on d. This improves upon an earlier result of Freiman, Heppes and Uhrin, and makes progress towards a conjecture
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The factorisation property of l∞(Xk) Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-12-10 RICHARD LECHNER; PAVLOS MOTAKIS; PAUL F.X. MÜLLER; THOMAS SCHLUMPRECHT
In this paper we consider the following problem: let Xk, be a Banach space with a normalised basis (e(k, j))j, whose biorthogonals are denoted by ${(e_{(k,j)}^*)_j}$ , for $k\in\N$ , let $Z=\ell^\infty(X_k:k\kin\N)$ be their l∞-sum, and let $T:Z\to Z$ be a bounded linear operator with a large diagonal, i.e., $$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*}$$ Under which
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A cancellation theorem for modules over integral group rings Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-11-27 JOHNNY NICHOLSON
A long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups G for which the integral group ring ℤG has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that ℤG has SFC provided at most one copy of the quaternions ℍ occurs in the Wedderburn decomposition of the real group ring ℝG.
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Directions in orbits of geometrically finite hyperbolic subgroups Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-11-23 CHRISTOPHER LUTSKO
We prove a theorem describing the limiting fine-scale statistics of orbits of a point in hyperbolic space under the action of a discrete subgroup. Similar results have been proved only in the lattice case with two recent infinite-volume exceptions by Zhang for Apollonian circle packings and certain Schottky groups. Our results hold for general Zariski dense, non-elementary, geometrically finite subgroups
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Zeros of the Epstein zeta function to the right of the critical line Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-11-17 YOUNESS LAMZOURI
Let E(s, Q) be the Epstein zeta function attached to a positive definite quadratic form of discriminant D < 0, such that h(D) ≥ 2, where h(D) is the class number of the imaginary quadratic field ${{\mathbb{Q}}(\sqrt D)}$ . We denote by NE(σ1, σ2, T) the number of zeros of E(s, Q) in the rectangle σ1 < Re(s) ≤ σ2 and T ≤ Im (s) ≤ 2T, where 1/2 < σ1 < σ2 < 1 are fixed real numbers. In this paper, we
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Non-torsion non-algebraic classes in the Brown–Peterson tower Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-10-16 MASAKI KAMEKO
Generalising the classical work of Atiyah and Hirzebruch on non-algebraic classes, recently Quick proved the existence of torsion non-algebraic elements in the Brown–Peterson tower. We construct non-torsion non-algebraic elements in the Brown–Peterson tower for the prime number 2.
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Popular differences for corners in Abelian groups Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-10-07 AARON BERGER
For a compact abelian group G, a corner in G × G is a triple of points (x, y), (x, y+d), (x+d, y). The classical corners theorem of Ajtai and Szemerédi implies that for every α > 0, there is some δ > 0 such that every subset A ⊂ G × G of density α contains a δ fraction of all corners in G × G, as x, y, d range over G. Recently, Mandache proved a “popular differences” version of this result in the finite
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Rational lines on cubic hypersurfaces Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-04-24 JULIA BRANDES; RAINER DIETMANN
We show that any smooth projective cubic hypersurface of dimension at least 29 over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to the second author and Wooley.
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Additive correlation and the inverse problem for the large sieve Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-07-09 BRANDON HANSON
Let A ⊆ [1, N] be a set of integers with |A| ≫ $\sqrt N$ . We show that if A avoids about p/2 residue classes modulo p for each prime p, then A must correlate additively with the squares S = {n2 : 1 ≤ n ≤ $\sqrt N$ }, in the sense that we have the additive energy estimate $$ E(A,S)\gg N\log N. $$ This is, in a sense, optimal.
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On the Saxl graph of a permutation group Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-08-08 TIMOTHY C. BURNESS; MICHAEL GIUDICI
Let G be a permutation group on a set Ω. A subset of Ω is a base for G if its pointwise stabiliser in G is trivial. In this paper we introduce and study an associated graph Σ(G), which we call the Saxl graph of G. The vertices of Σ(G) are the points of Ω, and two vertices are adjacent if they form a base for G. This graph encodes some interesting properties of the permutation group. We investigate
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Remarks on Wolff's inequality for hypersurfaces Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-09-06 SHAOMING GUO; CHANGKEUN OH
We run an iteration argument due to Pramanik and Seeger, to provide a proof of sharp decoupling inequalities for conical surfaces and for k-cones. These are extensions of results of Łaba and Pramanik to sharp exponents.
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On orbifold constructions associated with the Leech lattice vertex operator algebra Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-09-05 CHING HUNG LAM; HIROKI SHIMAKURA
In this paper, we study orbifold constructions associated with the Leech lattice vertex operator algebra. As an application, we prove that the structure of a strongly regular holomorphic vertex operator algebra of central charge 24 is uniquely determined by its weight one Lie algebra if the Lie algebra has the type A3,43A1,2, A4,52, D4,12A2,6, A6,7, A7,4A1,13, D5,8A1,2 or D6,5A1,12 by using the reverse
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Classification of the real discrete specialisations of the Burau representation of B3 Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-10-09 NANCY SCHERICH
This classification is found by analyzing the action of a normal subgroup of B3 as hyperbolic isometries. This paper gives an example of an unfaithful specialisation of the Burau representation on B4 that is faithful when restricted to B3, as well as examples of unfaithful specialisations of B3.
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The frame of smashing tensor-ideals Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-10-19 PAUL BALMER; HENNING KRAUSE; GREG STEVENSON
We prove that every flat tensor-idempotent in the module category Mod- of a tensor-triangulated category comes from a unique smashing ideal in . We deduce that the lattice of smashing ideals forms a frame.
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The structure of triple homomorphisms onto prime algebras Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-10-23 CHENG–KAI LIU
Triple homomorphisms on C*-algebras and JB*-triples have been studied in the literature. From the viewpoint of associative algebras, we characterise the structure of triple homomorphisms from an arbitrary ⋆-algebra onto a prime *-algebra. As an application, we prove that every triple homomorphism from a Banach ⋆-algebra onto a prime semisimple idempotent Banach *-algebra is continuous. The analogous
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Rational discrete first degree cohomology for totally disconnected locally compact groups Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-10-12 ILARIA CASTELLANO
It is well known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group G can be detected on the cohomology group H1(G,R[G]), where R is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group G the same information
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Sequences, modular forms and cellular integrals Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-10-11 DERMOT McCARTHY; ROBERT OSBURN; ARMIN STRAUB
It is well-known that the Apéry sequences which arise in the irrationality proofs for ζ(2) and ζ(3) satisfy many intriguing arithmetic properties and are related to the pth Fourier coefficients of modular forms. In this paper, we prove that the connection to modular forms persists for sequences associated to Brown's cellular integrals and state a general conjecture concerning supercongruences.
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A group version of stable regularity Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-10-24 G. CONANT; A. PILLAY; C. TERRY
We prove that, given ε > 0 and k ≥ 1, there is an integer n such that the following holds. Suppose G is a finite group and A ⊆ G is k-stable. Then there is a normal subgroup H ≤ G of index at most n, and a set Y ⊆ G, which is a union of cosets of H, such that |A △ Y| ≤ε|H|. It follows that, for any coset C of H, either |C ∩ A|≤ ε|H| or |C \ A| ≤ ε |H|. This qualitatively generalises recent work of
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PSP volume 168 issue 2 Cover and Front matter Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-02-11
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PSP volume 168 issue 2 Cover and Back matter Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-02-11
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Degeneration of Kummer surfaces Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-04-16 OTTO OVERKAMP
We prove that a Kummer surface defined over a complete strictly Henselian discretely valued field K of residue characteristic different from 2 admits a strict Kulikov model after finite base change. The Kulikov models we construct will be schemes, so our results imply that the semistable reduction conjecture is true for Kummer surfaces in this setup, even in the category of schemes. Our construction
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Tropical friezes and the index in higher homological algebra Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-03-16 PETER JØRGENSEN
Cluster categories and cluster algebras encode two dimensional structures. For instance, the Auslander–Reiten quiver of a cluster category can be drawn on a surface, and there is a class of cluster algebras determined by surfaces with marked points.
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Two polarised K3 surfaces associated to the same cubic fourfold Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-03-16 EMMA BRAKKEE
For infinitely many d, Hassett showed that special cubic fourfolds of discriminant d are related to polarised K3 surfaces of degree d via their Hodge structures. For half of the d, each associated K3 surface (S, L) canonically yields another one, (Sτ, Lτ). We prove that Sτ is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3, L, d/6). We also explain for which d the
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A combination theorem for combinatorially non-positively curved complexes of hyperbolic groups Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-03-09 ALEXANDRE MARTIN; DAMIAN OSAJDA
We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and C'(1/6) small cancellation polygonal complexes. Our proof involves constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the
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Monotone Lagrangians in of minimal Maslov number n + 1 Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-02-21 MOMCHIL KONSTANTINOV; JACK SMITH
We show that a monotone Lagrangian L in ${\mathbb{C}}{\mathbb{P}}^n$ of minimal Maslov number n + 1 is homeomorphic to a double quotient of a sphere, and thus homotopy equivalent to ${\mathbb{R}}{\mathbb{P}}^n$ . To prove this we use Zapolsky’s canonical pearl complex for L over ${\mathbb{Z}}$ , and twisted versions thereof, where the twisting is determined by connected covers of L. The main tool is
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Two closed orbits for non-degenerate Reeb flows Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-02-21 MIGUEL ABREU; JEAN GUTT; JUNGSOO KANG; LEONARDO MACARINI
We prove that every non-degenerate Reeb flow on a closed contact manifold M admitting a strong symplectic filling W with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic homology of W satisfies a mild condition. Under further assumptions, we establish the existence of two geometrically distinct closed orbits on any
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Short closed geodesics with self-intersections Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-01-24 VIVEKA ERLANDSSON; HUGO PARLIER
Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer k, we are interested in the set of all closed geodesics with at least k (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function
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Revisiting Leighton’s theorem with the Haar measure Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-01-13 DANIEL J. WOODHOUSE
Leighton’s graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton’s theorem that allows generalisations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen–Macura and Hagen–Touikan. To illustrate
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Moufang loops with nonnormal commutative centre Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2020-01-10 ALEXANDER N. GRISHKOV; ANDREI V. ZAVARNITSINE
We construct two infinite series of Moufang loops of exponent 3 whose commutative centre (i. e. the set of elements that commute with all elements of the loop) is not a normal subloop. In particular, we obtain examples of such loops of orders 38 and 311 one of which can be defined as the Moufang triplication of the free Burnside group B(3, 3).
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On the iteration of quasimeromorphic mappings Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-07-06 LUKE WARREN
The Fatou–Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to quasimeromorphic mappings with an essential singularity at infinity and at least one pole, constructing the Julia set for these maps. We show that this Julia set shares
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Krull's principal ideal theorem in non-Noetherian settings Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-08-08 BRUCE OLBERDING
Let P be a finitely generated ideal of a commutative ring R. Krull's principal ideal theorem states that if R is Noetherian and P is minimal over a principal ideal of R, then P has height at most one. Straightforward examples show that this assertion fails if R is not Noetherian. We consider what can be asserted in the non-Noetherian case in place of Krull's theorem.
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Geodesic X-ray tomography for piecewise constant functions on nontrapping manifolds Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-09-12 JOONAS ILMAVIRTA; JERE LEHTONEN; MIKKO SALO
We show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqueness result. Our proofs are elementary.
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Median and injective metric spaces Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-07-27 BRIAN H. BOWDITCH
We describe a construction which associates to any median metric space a pseudometric satisfying the binary intersection property for closed balls. Under certain conditions, this implies that the resulting space is, in fact, an injective metric space, bilipschitz equivalent to the original metric. In the course of doing this, we derive a few other facts about median metrics, and the geometry of CAT(0)
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Small sets containing any pattern Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-07-31 URSULA MOLTER; ALEXIA YAVICOLI
Given any dimension function h, we construct a perfect set E ⊆ ${\mathbb{R}}$ of zero h-Hausdorff measure, that contains any finite polynomial pattern.
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Regulator constants of integral representations of finite groups Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-09-05 ALEX TORZEWSKI
Let G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$ [G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of
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Interior of sums of planar sets and curves Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-09-05 KÁROLY SIMON; KRYSTAL TAYLOR
Recently, considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the interior (A + Γ)°, when Γ is a piecewise ${\mathcal C}^2$ curve and A ⊂ ℝ2. To begin, we give an example of a very large (full-measure, dense, Gδ) set A such that (A + S1)° = ∅, where S1 denotes the unit circle. This suggests that merely the size of A does not guarantee
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The distribution of consecutive prime biases and sums of sawtooth random variables Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-08-02 ROBERT J. LEMKE OLIVER; KANNAN SOUNDARARAJAN
In recent work, we considered the frequencies of patterns of consecutive primes (mod q) and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting
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Supercongruences for truncated hypergeometric series and p-adic gamma function Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-08-10 RUPAM BARMAN; NEELAM SAIKIA
We prove three more general supercongruences between truncated hypergeometric series and p-adic gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez--Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for 7F6 truncated hypergeometric
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On the weak forms of the 2-part of Birch and Swinnerton-Dyer conjecture Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2018-09-05 SHUAI ZHAI
In this paper, we investigate the weak forms of the 2-part of the conjecture of Birch and Swinnerton-Dyer, and prove a lower bound for the 2-adic valuation of the algebraic part of the central value of the complex L-series for the family of quadratic twists of all optimal elliptic curves over ${\mathbb Q}$ .
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PSP volume 168 issue 1 Cover and Front matter Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-12-18
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PSP volume 168 issue 1 Cover and Back matter Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-12-18
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On commensurability of right-angled Artin groups II: RAAGs defined by paths Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-12-12 MONTSERRAT CASALS–RUIZ; ILYA KAZACHKOV; ALEXANDER ZAKHAROV
In this paper we continue the study of right-angled Artin groups up to commensurability initiated in [CKZ]. We show that RAAGs defined by different paths of length greater than 3 are not commensurable. We also characterise which RAAGs defined by paths are commensurable to RAAGs defined by trees of diameter 4. More precisely, we show that a RAAG defined by a path of length n > 4 is commensurable to
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On square-free values of large polynomials over the rational function field Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-12-12 DAN CARMON; ALEXEI ENTIN
We investigate the density of square-free values of polynomials with large coefficients over the rational function field 𝔽q[t]. Some interesting questions answered as special cases of our results include the density of square-free polynomials in short intervals, and an asymptotic for the number of representations of a large polynomial N as a sum of a k-th power of a small polynomial and a square-free
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Escaping Fatou components of transcendental self-maps of the punctured plane Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-11-28 DAVID MARTÍ-PETE
We study the iteration of transcendental self-maps of $\,\mathbb{C}^*\!:=\mathbb{C}\setminus \{0\}$ , that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0,\infty\}$ in
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Counting subgraphs in fftp graphs with symmetry Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-11-27 YAGO ANTOLÍN
Following ideas that go back to Cannon, we show the rationality of various generating functions of growth sequences counting embeddings of convex subgraphs in locally-finite, vertex-transitive graphs with the (relative) falsification by fellow traveler property (fftp). In particular, we recover results of Cannon, of Epstein, Iano–Fletcher and Zwick, and of Calegari and Fujiwara. One of our applications
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Johnson–Levine homomorphisms and the tree reduction of the LMO functor Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-11-27 ANDERSON VERA
Let $\mathcal{M}$ denote the mapping class group of Σ, a compact connected oriented surface with one boundary component. The action of $\mathcal{M}$ on the nilpotent quotients of π1(Σ) allows to define the so-called Johnson filtration and the Johnson homomorphisms. J. Levine introduced a new filtration of $\mathcal{M}$ , called the Lagrangian filtration. He also introduced a version of the Johnson
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On the quotients of mapping class groups of surfaces by the Johnson subgroups Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-11-27 TOMÁŠ ZEMAN
We study quotients of mapping class groups ${\Gamma _{g,1}}$ of oriented surfaces with one boundary component by the subgroups ${{\cal I}_{g,1}}(k)$ in the Johnson filtrations, and we show that the stable classifying spaces ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(k))^ + }$ after plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates
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Rigidity dimension of algebras Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-11-26 HONGXING CHEN; MING FANG; OTTO KERNER; STEFFEN KOENIG; KUNIO YAMAGATA
A new homological dimension, called rigidity dimension, is introduced to measure the quality of resolutions of finite dimensional algebras (especially of infinite global dimension) by algebras of finite global dimension and big dominant dimension. Upper bounds of the dimension are established in terms of extensions and of Hochschild cohomology, and finiteness in general is derived from homological
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Lower Assouad Dimension of Measures and Regularity Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-11-22 KATHRYN E. HARE; SASCHA TROSCHEIT
In analogy with the lower Assouad dimensions of a set, we study the lower Assouad dimensions of a measure. As with the upper Assouad dimensions, the lower Assouad dimensions of a measure provide information about the extreme local behaviour of the measure. We study the connection with other dimensions and with regularity properties. In particular, the quasi-lower Assouad dimension is dominated by the
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Separation dimension and degree Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-11-22 ALEX SCOTT; DAVID R. WOOD
The separation dimension of a graph G is the minimum positive integer d for which there is an embedding of G into ℝd, such that every pair of disjoint edges are separated by some axis-parallel hyperplane. We prove a conjecture of Alon et al. [SIAM J. Discrete Math. 2015] by showing that every graph with maximum degree Δ has separation dimension less than 20Δ, which is best possible up to a constant
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Variation of the algebraic λ-invariant over a solvable extension Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-11-21 DANIEL DELBOURGO
Fix an odd prime p. Let $\mathcal{D}_n$ denote a non-abelian extension of a number field K such that $K\cap\mathbb{Q}(\mu_{p^{\infty}})=\mathbb{Q}, $ and whose Galois group has the form $ \text{Gal}\big(\mathcal{D}_n/K\big)\cong \big(\mathbb{Z}/p^{n'}\mathbb{Z}\big)^{\oplus g}\rtimes \big(\mathbb{Z}/p^n\mathbb{Z}\big)^{\times}\ $ where g > 0 and $0 \lt n'\leq n$ . Given a modular Galois representation
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Cube complexes and abelian subgroups of automorphism groups of RAAGs Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-11-20 BENJAMIN MILLARD; KAREN VOGTMANN
We construct free abelian subgroups of the group U(AΓ) of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group U(AΓ) was studied in [5] by constructing a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups
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KMS states on the C*-algebras of Fell bundles over groupoids Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-11-19 ZAHRA AFSAR; AIDAN SIMS
We consider fibrewise singly generated Fell bundles over étale groupoids. Given a continuous real-valued 1-cocycle on the groupoid, there is a natural dynamics on the cross-sectional algebra of the Fell bundle. We study the Kubo–Martin–Schwinger equilibrium states for this dynamics. Following work of Neshveyev on equilibrium states on groupoid C*-algebras, we describe the equilibrium states of the
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Maximal hyperbolic towers and weight in the theory of free groups Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-11-18 BENJAMIN BRÜCK
We show that in general for a given group the structure of a maximal hyperbolic tower over a free group is not canonical: we construct examples of groups having hyperbolic tower structures over free subgroups which have arbitrarily large ratios between their ranks. These groups have the same first order theory as non-abelian free groups and we use them to study the weight of types in this theory.
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Moments of the Dedekind zeta function and other non-primitive L-functions Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-11-15 WINSTON HEAP
We give a conjecture for the moments of the Dedekind zeta function of a Galois extension. This is achieved through the hybrid product method of Gonek, Hughes and Keating. The moments of the product over primes are evaluated using a theorem of Montgomery and Vaughan, whilst the moments of the product over zeros are conjectured using a heuristic method involving random matrix theory. The asymptotic formula
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Extensions of tensor categories by finite group fusion categories Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-11-11 SONIA NATALE; CIEM – CONICET
We study exact sequences of finite tensor categories of the form Rep G → 𝒞 → 𝒟, where G is a finite group. We show that, under suitable assumptions, there exists a group Γ and mutual actions by permutations ⊳ : Γ × G → G and ⊲ : Γ × G→ Γ that make (G, Γ) into matched pair of groups endowed with a natural crossed action on 𝒟 such that 𝒞 is equivalent to a certain associated crossed extension 𝒟(G
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Vinogradov’s three primes theorem with primes having given primitive roots Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-11-05 C. FREI; P. KOYMANS; E. SOFOS
The first purpose of our paper is to show how Hooley’s celebrated method leading to his conditional proof of the Artin conjecture on primitive roots can be combined with the Hardy–Littlewood circle method. We do so by studying the number of representations of an odd integer as a sum of three primes, all of which have prescribed primitive roots. The second purpose is to analyse the singular series.
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Bisecting measures with hyperplane arrangements Math. Proc. Camb. Philos. Soc. (IF 0.741) Pub Date : 2019-10-31 ALFREDO HUBARD; ROMAN KARASEV
We show that provided that n is a power of two, any nD measures in ℝn can be bisected by an arrangement of D hyperplanes.