arXiv research

Loops linked to p-adic manifolds studied.

Develops Weil bundles over \( p \)-adic manifolds for arithmetic geometry.

A conceptual framework for cluster analysis from the viewpoint of p-adic geometry is introduced by describing the space of all dendrograms for n datapoints and relating it to the moduli space of p-adic Riemannian spheres with punctures using a method recently applied by Murtagh (2004b). This method embeds a dendrogram …

Dendrograms used in data analysis are ultrametric spaces, hence objects of nonarchimedean geometry. It is known that there exist $p$-adic representation of dendrograms. Completed by a point at infinity, they can be viewed as subtrees of the Bruhat-Tits tree associated to the $p$-adic projective line. The implications a…

We give a brief overview of the theory of complex dimensions of real (archimedean) fractal strings via an illustrative example, the ordinary Cantor string, and a detailed survey of the theory of p-adic (nonarchimedean) fractal strings and their complex dimensions. Moreover, we present an explicit volume formula for the…

Study of $p$-adic simplicial volumes and their properties.

The paper describes p-adic analogs of hyperbolic discs and their geometric properties.

The Yokonuma-Hecke algebras are quotients of the modular framed braid group and they support Markov traces. In this paper, which is sequel to Juyumaya and Lambropoulou (2007), we explore further the structures of the $p$-adic framed braids and the $p$-adic Yokonuma-Hecke algebras constructed in Juyumaya and Lambropoulo…

Theory developed for $p$-adic Mahler measure applied to $\mathbb{Z}$-covers of links.

The study examines compact spaces resolvable by p-adic actions.

This paper reformulates the $p$-adic Littlewood Conjecture using infinite loops.

Simplified construction of p-adic transformation group action.

Triangulates semi-algebraic sets over p-adic fields, proving similar results to reals.

The paper studies $p$-adic limits of class numbers in $\mathbb{Z}_p$-extensions and covers.

Author simplifies and generalizes p-adic integer action construction.

Geometric zeta functions of Ihara and Hashimoto are generalized to higher rank. The $p$-adic version of the Patterson conjecture is proven.

A new multiagent model of the stock market is formulated that contains four states in which the agents may be located. Next, the model is reformulated in the language of the functional integral containing fluctuations of prices and quantities of cash flows. It is shown that in the functional integral of that type descr…

In this paper we define the $p$-adic framed braid group ${\mathcal F}_{\infty,n}$, arising as the inverse limit of the modular framed braids and we give topological generators for ${\mathcal F}_{\infty, n}$. We also give geometric interpretations for the $p$-adic framed braids. We then construct a $p$-adic Yokonuma-Hec…

Develops arithmetic PDE geometry using Fermat quotients.

We study the asymptotic growth of Betti numbers in tower of finite covers and provide simple proofs of approximation results, which were previously obtained by Calegari-Emerton, in the generality of arbitrary p-adic analytic towers of covers. Further, we also obtain partial results about arbitrary pro-$p$ towers.

In this paper we represent the classical braids in the Yokonuma--Hecke and the adelic Yokonuma--Hecke algebras. More precisely, we define the completion of the framed braid group and we introduce the adelic Yokonuma--Hecke algebras, in analogy to the $p$--adic framed braids and the $p$--adic Yokonuma--Hecke algebras in…

We suggest to use the Hall-Littlewood version of Rosso-Jones formula to define the germs of $p$-adic HOMFLY-PT polynomials for torus knots $[m,n]$, which possess at least the $[m,n] \longleftrightarrow [n,m]$ topological invariance. This calls for generalizations to other knot families and is a challenge for several br…

The purpose of this article is to present a survey of our recent results on length commensurable and isospectral locally symmetric spaces. The geometric questions led us to the notion of "weak commensurability" of two Zariski-dense subgroups in a semi-simple Lie group. We have shown that for arithmetic subgroups, weak …

Abstract: Necessary condition for critical sets in 3D space.

Constructs operations on stable moduli spaces to compare manifold cohomology.

Levine defined the rational algebraic knot concordance group and proved that each nontrivial element is of order two, of order four, or of infinite order. The determination of the order of an element depends on a p-adic analysis for all primes p. Here we develop effective means to determine the order of any element tha…

Survey of Weber's class number problem and related topics.

Finite actions of lattices on manifolds proven for certain groups.

New example solves topological dynamics problem.

The paper contains a combinatorial theorem (the sequence of Newton polygons of a reccurent sequence of polynomials is quasi-linear) and two applications of it in classical and quantum topology, namely in the behavior of the $A$-polynomial and a fixed quantum invariant (such as the Jones polynomial) under filling. Our c…

Let p and l be two distinct prime numbers and let G be a group. We study the asymptotic behaviour of the mod-l Betti numbers in p-adic analytic towers of finite index subgroups. If X is a finite l-group of automorphisms of G, our main theorem allows to lift lower bounds for the mod-l cohomology growth in the fixed poin…

We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry group that is virtually a product of a solvable real Lie group and a solvable p-adic…

Arithmetic topology connects surface and $p$-adic field studies, enabling new insights into Galois groups.

Anabelian geometry reformulated using Hodge theory for hyperbolic curves.

Lecture notes on homology growth and approximation theorems.

Magnitude of manifolds linked to Riesz energies and beta functions.

The p-adic theory of the stock market is presented. It is shown that the price dynamics is very naturally described by the adelic function. The procedure of derivation of the functional integral formulation of adelic type is derived from microscopic models using generalized supercoherent states.

Classifies limits of groups of involutions in SL(2,F) over local fields.

Proves section conjecture for curves and surface bundles over various fields.

Arithmetic study of knots connects homology and SL2 representations.

Study shows non-existence of almost complex structures on certain sphere bundles over complex projective spaces.

For a link $L$ in the 3-sphere and for a prime $p$, we express the $p$-primary information on the first homology group of $p^{m}$-fold branched covers of $L$ in terms of its $p$-adic Milnor higher linking invariants, using the completed Alexander module of the pro-$p$ completion of the link group of $L$.

This is the topological part of two papers on the cohomology of Kaehler groups. In this paper we show that if a linear duality group of dimension larger than 6 is the fundamental group of a compact Kaehler manifold then its second or its fourth Betti number is non-zero. As a corollary a cocompact p-adic lattice of rank…

We study the spectral gap of the Erdős--Rényi random graph through the connectivity threshold. In particular, we show that for any fixed $δ> 0$ if $p \ge \frac{(1/2 + δ) \log n}{n},$ then the normalized graph Laplacian of an Erdős--Rényi graph has all of its nonzero eigenvalues tightly concentrated around $1$. We est…

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal …

We consider compact, aspherical solenoids obtained as the inverse limit of a system of CW~complexes and covering maps. This includes $P$-adic solenoids, as well as the universal hyperbolic solenoid of Teichmüller theory. Using ideas from shape theory, we classify maps between such solenoids up to homotopy, and we prove…

This paper provides a topological interpretation for number theoretic properties of quantum invariants of 3-manifolds. In particular, it is shown that the p-adic valuation of the quantum SO(3)-invariant of a 3-manifold M, for odd primes p, is bounded below by a linear function of the mod p first betti number of M. Shar…

The Hilbert-Smith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is given. The motivation is work of Cernavskii (``Finite-to-one mappings of manifolds'',…