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Mathematical Proceedings of the Cambridge Philosophical Society
Journal Prestige (SJR): 1.086

Citation Impact (citeScore): 1
Number of Followers: 2

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ISSN (Print) 0305-0041 - ISSN (Online) 1469-8064
Published by Cambridge University Press

[396 journals]

PSP volume 170 issue 3 Cover and Front matter
- PubDate: 2021-05-01T00:00:00.000Z
  DOI: 10.1017/S0305004121000347
  Issue No: Vol. 170, No. 3 (2021)
PSP volume 170 issue 3 Cover and Back matter
- PubDate: 2021-05-01T00:00:00.000Z
  DOI: 10.1017/S0305004121000359
  Issue No: Vol. 170, No. 3 (2021)
A combination theorem for combinatorially non-positively curved complexes
of hyperbolic groups
- Authors: ALEXANDRE MARTIN; DAMIAN OSAJDA
  Pages: 445 - 477
  Abstract: 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 action on the boundary in order to use Bowditch’s characterisation of hyperbolicity. A key ingredient is the introduction of a combinatorial property that implies a weak form of non-positive curvature, and which holds for large classes of complexes.As an application, we study the hyperbolicity of groups obtained by small cancellation over a graph of hyperbolic groups.
  PubDate: 2021-05-01T00:00:00.000Z
  DOI: 10.1017/S0305004119000446
  Issue No: Vol. 170, No. 3 (2021)
Maximal hyperbolic towers and weight in the theory of free groups
- Authors: BENJAMIN BRÜCK
  Pages: 479 - 498
  Abstract: 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.
  PubDate: 2021-05-01T00:00:00.000Z
  DOI: 10.1017/S0305004119000483
  Issue No: Vol. 170, No. 3 (2021)
Variation of the algebraic λ-invariant over a solvable extension
- Authors: DANIEL DELBOURGO
  Pages: 499 - 521
  Abstract: 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 $\overline{\rho}:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F})$ which is p-ordinary and also p-distinguished, we shall write $\mathcal{H}(\overline{\rho})$ for the associated Hida family. Using Greenberg’s notion of Selmer atoms, we prove an exact formula for the algebraic λ-invariant\begin{equation}\lambda^{\text{alg}}_{\mathcal{D}_n}(f) \;=\; \text{the number of zeroes of }\text{char}_{\Lambda}\big(\text{Sel}_{\mathcal{D}_n^{\text{cy}}}\big(f\big)^{\wedge}\big)\end{equation}at all $f\in\mathcal{H}(\overline{\rho})$, under the assumption $\mu^{\text{alg}}_{K(\mu_p)}(f_0)=0$ for at least one form f0. We can then easily deduce that $\lambda^{\text{alg}}_{\mathcal{D}_n}(f)$ is constant along branches of
  PubDate: 2021-05-01T00:00:00.000Z
  DOI: 10.1017/S0305004119000495
  Issue No: Vol. 170, No. 3 (2021)
Cube complexes and abelian subgroups of automorphism groups of RAAGs
- Authors: BENJAMIN MILLARD; KAREN VOGTMANN
  Pages: 523 - 547
  Abstract: 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 are equal to the dimensions of principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bounds agree. In many cases when the principal cubes are not of maximal dimension we show there is an invariant contractible subcomplex of strictly lower dimension.
  PubDate: 2021-05-01T00:00:00.000Z
  DOI: 10.1017/S0305004119000501
  Issue No: Vol. 170, No. 3 (2021)
Separation dimension and degree
- Authors: ALEX SCOTT; DAVID R. WOOD
  Pages: 549 - 558
  Abstract: 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 factor. We also prove that graphs with separation dimension 3 have bounded average degree and bounded chromatic number, partially resolving an open problem by Alon et al. [J. Graph Theory 2018].
  PubDate: 2021-05-01T00:00:00.000Z
  DOI: 10.1017/S0305004119000525
  Issue No: Vol. 170, No. 3 (2021)
On commensurability of right-angled Artin groups II: RAAGs defined by
paths
- Authors: MONTSERRAT CASALS–RUIZ; ILYA KAZACHKOV, ALEXANDER ZAKHAROV
  Pages: 559 - 608
  Abstract: 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 a RAAG defined by a tree of diameter 4 if and only if n ≡ 2 (mod 4). These results follow from the connection that we establish between the classification of RAAGs up to commensurability and linear integer-programming.
  PubDate: 2021-05-01T00:00:00.000Z
  DOI: 10.1017/S0305004119000537
  Issue No: Vol. 170, No. 3 (2021)
Moufang loops with nonnormal commutative centre
- Authors: ALEXANDER N. GRISHKOV; ANDREI V. ZAVARNITSINE
  Pages: 609 - 614
  Abstract: 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).
  PubDate: 2021-05-01T00:00:00.000Z
  DOI: 10.1017/S0305004119000549
  Issue No: Vol. 170, No. 3 (2021)
Revisiting Leighton’s theorem with the Haar measure
- Authors: DANIEL J. WOODHOUSE
  Pages: 615 - 623
  Abstract: 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 the potential for future applications, we give a quasi-isometric rigidity result for a family of cyclic doubles of free groups.
  PubDate: 2021-05-01T00:00:00.000Z
  DOI: 10.1017/S0305004119000550
  Issue No: Vol. 170, No. 3 (2021)
Two closed orbits for non-degenerate Reeb flows
- Authors: MIGUEL ABREU; JEAN GUTT, JUNGSOO KANG, LEONARDO MACARINI
  Pages: 625 - 660
  Abstract: 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 contact finite quotient of M. Several examples of such contact manifolds are provided, like displaceable ones, unit cosphere bundles, prequantisation circle bundles, Brieskorn spheres and toric contact manifolds. We also show that this condition on the equivariant symplectic homology is preserved by boundary connected sums of Liouville domains. As a byproduct of one of our applications, we prove a sort of Lusternik–Fet theorem for Reeb flows on the unit cosphere bundle of not rationally aspherical manifolds satisfying suitable additional assumptions.
  PubDate: 2021-05-01T00:00:00.000Z
  DOI: 10.1017/S0305004120000018
  Issue No: Vol. 170, No. 3 (2021)