Mathematical Proceedings of the Cambridge Philosophical Society: Volume 169 - Issue 2

Embedding right-angled Artin groups into Brin–Thompson groups
Part of:
- Special aspects of infinite or finite groups
- Structure and classification of infinite or finite groups
JAMES BELK, COLLIN BLEAK, FRANCESCO MATUCCI
Published online by Cambridge University Press: 23 April 2019, pp. 225-229
- Article
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
View abstract

We prove that every finitely-generated right-angled Artin group embeds into some Brin–Thompson group nV. It follows that any virtually special group can be embedded into some nV, a class that includes surface groups, all finitely-generated Coxeter groups, and many one-ended hyperbolic groups.

The multistep homology of the simplex and representations of symmetric groups
Part of:
- Representation theory of groups
- Homological algebra
MARK WILDON
Published online by Cambridge University Press: 20 June 2019, pp. 231-253
- Article
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
View abstract

The symmetric group on a set acts transitively on the set of its subsets of a fixed size. We define homomorphisms between the corresponding permutation modules, defined over a field of characteristic two, which generalize the boundary maps from simplicial homology. The main results determine when these chain complexes are exact and when they are split exact. As a corollary we obtain a new explicit construction of the basic spin modules for the symmetric group.

Types of embedded graphs and their Tutte polynomials
Part of:
- Graph theory
STEPHEN HUGGETT, IAIN MOFFATT
Published online by Cambridge University Press: 02 July 2019, pp. 255-297
- Article
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
View abstract

We take an elementary and systematic approach to the problem of extending the Tutte polynomial to the setting of embedded graphs. Four notions of embedded graphs arise naturally when considering deletion and contraction operations on graphs on surfaces. We give a description of each class in terms of coloured ribbon graphs. We then identify a universal deletion-contraction invariant (i.e., a ‘Tutte polynomial’) for each class. We relate these to graph polynomials in the literature, including the Bollobás–Riordan, Krushkal and Las Vergnas polynomials, and give state-sum formulations, duality relations, deleton-contraction relations, and quasi-tree expansions for each of them.

Recurrence of quadratic differentials for harmonic measure
Part of:
VAIBHAV GADRE, JOSEPH MAHER
Published online by Cambridge University Press: 25 June 2019, pp. 299-305
- Article
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
View abstract

We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum of quadratic differentials. We show that a Teichmüller geodesic typical with respect to the harmonic measure for such random walks, is recurrent to the thick part of the principal stratum. As a consequence, the vertical foliation of such a random Teichmüller geodesic has no saddle connections.

A comparison principle for convolution measures with applications
Part of:
DIOGO OLIVEIRA E SILVA, RENÉ QUILODRÁN
Published online by Cambridge University Press: 28 June 2019, pp. 307-322
- Article
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
View abstract

We establish the general form of a geometric comparison principle for n-fold convolutions of certain singular measures in ℝd which holds for arbitrary n and d. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in the companion paper [3].

Proof of the Brown–Erdős–Sós conjecture in groups
Part of:
- Graph theory
- Abelian groups
RAJKO NENADOV, BENNY SUDAKOV, MYKHAYLO TYOMKYN
Published online by Cambridge University Press: 04 July 2019, pp. 323-333
- Article
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
View abstract

The conjecture of Brown, Erdős and Sós from 1973 states that, for any k ≥ 3, if a 3-uniform hypergraph H with n vertices does not contain a set of k +3 vertices spanning at least k edges then it has o(n2) edges. The case k = 3 of this conjecture is the celebrated (6, 3)-theorem of Ruzsa and Szemerédi which implies Roth’s theorem on 3-term arithmetic progressions in dense sets of integers. Solymosi observed that, in order to prove the conjecture, one can assume that H consists of triples (a, b, ab) of some finite quasigroup Γ. Since this problem remains open for all k ≥ 4, he further proposed to study triple systems coming from finite groups. In this case he proved that the conjecture holds also for k = 4. Here we completely resolve the Brown–Erdős–Sós conjecture for all finite groups and values of k. Moreover, we prove that the hypergraphs coming from groups contain sets of size $\Theta (\sqrt k )$ which span k edges. This is best possible and goes far beyond the conjecture.

Tight closure of powers of ideals and tight hilbert polynomials
Part of:
KRITI GOEL, J. K. VERMA, VIVEK MUKUNDAN
Published online by Cambridge University Press: 12 July 2019, pp. 335-355
- Article
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
View abstract

Let (R, ) be an analytically unramified local ring of positive prime characteristic p. For an ideal I, let I* denote its tight closure. We introduce the tight Hilbert function $$H_I^*\left( n \right) = \Im \left( {R/\left( {{I^n}} \right)*} \right)$$ and the corresponding tight Hilbert polynomial $$P_I^*\left( n \right)$$, where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of $$P_I^*\left( n \right)$$. We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.

Open manifolds with non-homeomorphic positively curved souls
Part of:
DAVID GONZÁLEZ-ÁLVARO, MARCUS ZIBROWIUS
Published online by Cambridge University Press: 11 July 2019, pp. 357-376
- Article
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
View abstract

We extend two known existence results to simply connected manifolds with positive sectional curvature: we show that there exist pairs of simply connected positively-curved manifolds that are tangentially homotopy equivalent but not homeomorphic, and we deduce that an open manifold may admit a pair of non-homeomorphic simply connected and positively-curved souls. Examples of such pairs are given by explicit pairs of Eschenburg spaces. To deduce the second statement from the first, we extend our earlier work on the stable converse soul question and show that it has a positive answer for a class of spaces that includes all Eschenburg spaces.

Hilbert’s 16th problem on a period annulus and Nash space of arcs
Part of:
- Qualitative theory
- Birational geometry
JEAN–PIERRE FRANÇOISE, LUBOMIR GAVRILOV, DONGMEI XIAO
Published online by Cambridge University Press: 12 July 2019, pp. 377-409
- Article
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
View abstract

This paper introduces an algebro-geometric setting for the space of bifurcation functions involved in the local Hilbert’s 16th problem on a period annulus. Each possible bifurcation function is in one-to-one correspondence with a point in the exceptional divisor E of the canonical blow-up BI ℂn of the Bautin ideal I. In this setting, the notion of essential perturbation, first proposed by Iliev, is defined via irreducible components of the Nash space of arcs Arc(BI ℂn, E). The example of planar quadratic vector fields in the Kapteyn normal form is further discussed.

On the residual and profinite closures of commensurated subgroups
Part of:
- Structure and classification of infinite or finite groups
PIERRE–EMMANUEL CAPRACE, PETER H. KROPHOLLER, COLIN D. REID, PHILLIP WESOLEK
Published online by Cambridge University Press: 30 July 2019, pp. 411-432
- Article
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
View abstract

The residual closure of a subgroup H of a group G is the intersection of all virtually normal subgroups of G containing H. We show that if G is generated by finitely many cosets of H and if H is commensurated, then the residual closure of H in G is virtually normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.

PSP volume 169 issue 2 Cover and Front matter
Published online by Cambridge University Press: 18 August 2020, pp. f1-f2
- Article
- - You have access
- PDF
- Export citation

PSP volume 169 issue 2 Cover and Back matter
Published online by Cambridge University Press: 18 August 2020, pp. b1-b2
- Article
- - You have access
- PDF
- Export citation

Mathematical Proceedings of the Cambridge Philosophical Society

Refine listing

Actions for selected content:

Volume 169 - Issue 2 - September 2020

Research Article

Embedding right-angled Artin groups into Brin–Thompson groups

The multistep homology of the simplex and representations of symmetric groups

Types of embedded graphs and their Tutte polynomials

Recurrence of quadratic differentials for harmonic measure

A comparison principle for convolution measures with applications

Proof of the Brown–Erdős–Sós conjecture in groups

Tight closure of powers of ideals and tight hilbert polynomials

Open manifolds with non-homeomorphic positively curved souls

Hilbert’s 16th problem on a period annulus and Nash space of arcs

On the residual and profinite closures of commensurated subgroups

Front Cover (OFC, IFC) and matter

PSP volume 169 issue 2 Cover and Front matter

Back Cover (OBC, IBC) and matter

PSP volume 169 issue 2 Cover and Back matter

Mathematical Proceedings of the Cambridge Philosophical Society

Refine listing

Actions for selected content:

Save Search

Volume 169 - Issue 2 - September 2020

Research Article

Front Cover (OFC, IFC) and matter

Back Cover (OBC, IBC) and matter