On Cayley graphs of virtually free groups

In 1985, Dunwoody showed that finitely presentable groups are accessible. Dunwoody's result was used to show that context-free groups, groups quasi-isometric to trees or finitely presentable groups of asymptotic dimension 1 are virtually free. Using another theorem of Dunwoody of 1979, we study when a group is virtually free in terms of its Cayley graph and we obtain new proofs of the mentioned results and other previously depending on them

Coin.AI: A Proof-of-Useful-Work Scheme for Blockchain-based Distributed Deep Learning

One decade ago, Bitcoin was introduced, becoming the first cryptocurrency and establishing the concept of "blockchain" as a distributed ledger. As of today, there are many different implementations of cryptocurrencies working over a blockchain, with different approaches and philosophies. However, many of them share one common feature: they require proof-of-work to support the generation of blocks (mining) and, eventually, the generation of money. This proof-of-work scheme often consists in the resolution of a cryptography problem, most commonly breaking a hash value, which can only be achieved through brute-force. The main drawback of proof-of-work is that it requires ridiculously large amounts of energy which do not have any useful outcome beyond supporting the currency. In this paper, we present a theoretical proposal that introduces a proof-of-useful-work scheme to support a cryptocurrency running over a blockchain, which we named Coin.AI. In this system, the mining scheme requires training deep learning models, and a block is only mined when the performance of such model exceeds a threshold. The distributed system allows for nodes to verify the models delivered by miners in an easy way (certainly much more efficiently than the mining process itself), determining when a block is to be generated. Additionally, this paper presents a proof-of-storage scheme for rewarding users that provide storage for the deep learning models, as well as a theoretical dissertation on how the mechanics of the system could be articulated with the ultimate goal of democratizing access to artificial intelligence.Comment: 17 pages, 5 figure

The Haagerup property is stable under graph products

The Haagerup property, which is a strong converse of Kazhdan's property $(T)$, has translations and applications in various fields of mathematics such as representation theory, harmonic analysis, operator K-theory and so on. Moreover, this group property implies the Baum-Connes conjecture and related Novikov conjecture. The Haagerup property is not preserved under arbitrary group extensions and amalgamated free products over infinite groups, but it is preserved under wreath products and amalgamated free products over finite groups. In this paper, we show that it is also preserved under graph products. We moreover give bounds on the equivariant and non-equivariant $L_p$-compressions of a graph product in terms of the corresponding compressions of the vertex groups. Finally, we give an upper bound on the asymptotic dimension in terms of the asymptotic dimensions of the vertex groups. This generalizes a result from Dranishnikov on the asymptotic dimension of right-angled Coxeter groups.Comment: 20 pages, v3 minor change

Tits alternatives for graph products

We discuss various types of Tits Alternative for subgroups of graph products of groups, and prove that, under some natural conditions, a graph product of groups satisfies a given form of Tits Alternative if and only if each vertex group satisfies this alternative. As a corollary, we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large. As another corollary, we prove that every non-abelian subgroup of a right angled Artin group has an epimorphism onto the free group of rank 2. In the course of the paper we develop the theory of parabolic subgroups, which allows to describe the structure of subgroups of graph products that contain no non-abelian free subgroups. We also obtain a number of results regarding the stability of some group properties under taking graph products