The Cayley isomorphism property for Cayley maps

In this paper we study finite groups which have Cayley isomorphism property with respect to Cayley maps, CIM-groups for a brief. We show that the structure of the CIM-groups is very restricted. It is described in Theorem~\ref{111015a} where a short list of possible candidates for CIM-groups is given. Theorem~\ref{111015c} provides concrete examples of infinite series of CIM-groups

Approximating Cayley diagrams versus Cayley graphs

We construct a sequence of finite graphs that weakly converge to a Cayley graph, but there is no labelling of the edges that would converge to the corresponding Cayley diagram. A similar construction is used to give graph sequences that converge to the same limit, and such that a spanning tree in one of them has a limit that is not approximable by any subgraph of the other. We give an example where this subtree is a Hamiltonian cycle, but convergence is meant in a stronger sense. These latter are related to whether having a Hamiltonian cycle is a testable graph property.Comment: 8 pages, 1 figur

Amalgamation and Symmetry: From Local to Global Consistency in The Finite

Amalgamation patterns are specified by a finite collection of finite template structures together with a collection of partial isomorphisms between pairs of these. The template structures specify the local isomorphism types that occur in the desired amalgams; the partial isomorphisms specify local amalgamation requirements between pairs of templates. A realisation is a globally consistent solution to the locally consistent specification of this amalgamation problem. This is a single structure equipped with an atlas of distinguished substructures associated with the template structures in such a manner that their overlaps realise precisely the identifications induced by the local amalgamation requirements. We present a generic construction of finite realisations of amalgamation patterns. Our construction is based on natural reduced products with suitable groupoids. The resulting realisations are generic in the sense that they can be made to preserve all symmetries inherent in the specification, and can be made to be universal w.r.t. to local homomorphisms up to any specified size. As key applications of the main construction we discuss finite hypergraph coverings of specified levels of acyclicity and a new route to the lifting of local symmetries to global automorphisms in finite structures in the style of Herwig-Lascar extension properties for partial automorphisms.Comment: A mistake in the proposed construction from [arXiv:1211.5656], cited in Theorem 3.21, was discovered by Julian Bitterlich. This version relies on the new approach to this construction as presented in the new version of [arXiv:1806.08664

Pseudo-modularity and Iwasawa theory

We prove, assuming Greenberg's conjecture, that the ordinary eigencurve is Gorenstein at an intersection point between the Eisenstein family and the cuspidal locus. As a corollary, we obtain new results on Sharifi's conjecture. This result is achieved by constructing a universal ordinary pseudodeformation ring and proving an $R = \mathbb T$ result.Comment: Changes to section 5.9; typos corrected. To appear in Amer. J. Math. 54 page

A classification of nilpotent 3-BCI groups

Given a finite group $G$ and a subset $S\subseteq G,$ the bi-Cayley graph \bcay(G,S) is the graph whose vertex set is $G \times \{0,1\}$ and edge set is $\{\{(x,0),(s x,1)\} : x \in G, s\in S \}$. A bi-Cayley graph \bcay(G,S) is called a BCI-graph if for any bi-Cayley graph \bcay(G,T), \bcay(G,S) \cong \bcay(G,T) implies that $T = g S^\alpha$ for some $g \in G$ and \alpha \in \aut(G). A group $G$ is called an $m$-BCI-group if all bi-Cayley graphs of $G$ of valency at most $m$ are BCI-graphs.In this paper we prove that, a finite nilpotent group is a 3-BCI-group if and only if it is in the form $U \times V,$ where $U$ is a homocyclic group of odd order, and $V$ is trivial or one of the groups $\Z_{2^r},$ $\Z_2^r$ and \Q_8

The topology of the minimal regular cover of the Archimedean tessellations

In this article we determine, for an infinite family of maps on the plane, the topology of the surface on which the minimal regular covering occurs. This infinite family includes all Archimedean maps.Comment: 21 pages, 9 figure

Elementary Abelian p-groups of rank 2p+3 are not CI-groups

For every prime $p > 2$ we exhibit a Cayley graph of $\mathbb{Z}_p^{2p+3}$ which is not a CI-graph. This proves that an elementary Abelian $p$-group of rank greater than or equal to $2p+3$ is not a CI-group. The proof is elementary and uses only multivariate polynomials and basic tools of linear algebra. Moreover, we apply our technique to give a uniform explanation for the recent works concerning the bound.Comment: 11 page

Algebraic families of Galois representations and potentially semi-stable pseudodeformation rings

We construct and study the moduli of continuous representations of a profinite group with integral $p$-adic coefficients. We present this moduli space over the moduli space of continuous pseudorepresentations and show that this morphism is algebraizable. When this profinite group is the absolute Galois group of a $p$-adic local field, we show that these moduli spaces admit Zariski-closed loci cutting out Galois representations that are potentially semi-stable with bounded Hodge-Tate weights and a given Hodge and Galois type. As a consequence, we show that these loci descend to the universal deformation ring of the corresponding pseudorepresentation.Comment: Numbering changed and typos corrected to match published version. 59 page

Finite Groupoids, Finite Coverings and Symmetries in Finite Structures

We propose a novel construction of finite hypergraphs and relational structures that is based on reduced products with Cayley graphs of groupoids. To this end we construct groupoids whose Cayley graphs have large girth not just in the usual sense, but with respect to a discounted distance measure that contracts arbitrarily long sequences of edges within the same sub-groupoid (coset) and only counts transitions between cosets. Reduced products with such groupoids are sufficiently generic to be applicable to various constructions that are specified in terms of local glueing operations and require global finite closure. We here examine hypergraph coverings and extension tasks that lift local symmetries to global automorphisms.Comment: This paper extends and supersedes earlier expositions in LICS 2013 and arXiv:1211.5656. Version (v2) of this paper fixes a false claim in Lemma 2.9 of the original version. Version (v4) eliminates confusion around "covering of A" vs "realisation of H(A)" (Definition 3.14 and adaptation of Lemma 3.16) and corrects a mistake in the "excursion" on Herwig's thm in section 4.

The fundamental group and covering spaces

These lecture notes from a first course in algebraic topology use the fundamental group and orbit categories to classify covering spaces.Comment: 31 page