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Gabor Elek

Publications and source records attributed to Gabor Elek.

34 records · Page 2Linked to original sources
arXiv2009

Borel oracles. An analytical approach to constant-time algorithms

Nguyen and Onak constructed the first constant-time algorithm for the approximation of the size of the maximum matching in bounded degree graphs. The Borel oracle machinery is a tool that can be used to convert some statements in Borel graph theory to theorems in the field of constant-time algorithms. In this paper we illustrate the power of this tool to prove the existence of the above mentioned constant-time approximation algorithm.

math.CO↗
arXiv2009

Parameter testing with bounded degree graphs of subexponential growth

Parameter testing algorithms are using constant number of queries to estimate the value of a certain parameter of a very large finite graph. It is well-known that graph parameters such as the independence ratio or the edit-distance from 3-colorability are not testable in bounded degree graphs. We prove, however, that these and several other interesting graph parameters are testable in bounded degree graphs of subexponential growth.

math.CO↗
arXiv2008

On the limit of large girth graph sequences

We prove that any involution-invariant probability measure on the space of trees with maximum degrees at most d arises as the local limit of a convergent large girth graph sequence. This answers a question of Bollobas and Riordan.

math.CO↗
arXiv2007

Non-abelian free groups admit non-essentially free actions on rooted trees

We show that every countable non-abelian free group $Γ$ admits a spherically transitive action on a rooted tree $T$ such that the action of $Γ$ on the boundary of $T$ is not essentially free. This reproves a result of Bergeron and Gaboriau. The existence of such an action answers a question of Grigorchuk, Nekrashevich and Sushchanskii.

math.GR↗
arXiv2006

Aperiodic order, integrated density of states and the continuous algebras of John von Neumann

Lenz and Stollmann recently proved the existence of the integrated density of states in the sense of uniform convergence of the distributions for certain operators with aperiodic order. The goal of this paper is to establish a relation between aperiodic order, uniform spectral convergence and the continuous algebras invented by John von Neumann. We illustrate the technique by proving the uniform spectral convergence for random Schodinger operators on lattices with finite site probabilities, percolation Hamiltonians and for the pattern-invariant operators of self-similar graphs.

math-ph↗
arXiv2005

On limits of finite graphs

We prove that for any weakly convergent sequence of finite graphs with bounded vertex degrees, there exists a topological limit graphing.

math.CO↗
arXiv2005

The Strong Approximation Conjecture holds for amenable groups

We prove that the approximation conjecture of Luck holds for all amenable groups in the complex group algebra case. This result was previously proved by Dodziuk, Linnell, Mathai, Schick and Yates under the assumption that the group is torsion-free elementary amenable.

math.FA↗
arXiv2005

The Ends of Algebras

We introduce the notion of ends for algebras. The definition is analogous to the one in geometric group theory. We establish some relations to growth conditions and cyclic cohomology.

math.RA↗
arXiv2002

The amenability of affine algebras

We introduce the notion of amenability for affine algebras. We characterize amenability by Folner-sequences, paradoxicality and the existence of finitely invariant dimension-measures. Then we extend the results of Rowen on ranks, from affine algebras of subexponential growth to amenable affine algebras.

math.RA↗
arXiv1998

The Euler characteristic of discrete groups and Yuzvinskii's entropy addition formula

The notion of topological entropy is originally defined for a single action. Later it was extended by Kieffer for arbitrary discrete amenable groups. Recently Friedland defined topological entropy for any discrete group actions amenable or not. Lind, Schmidt and Ward extended Yuzvinskii's addition formula from single action to abelian group actions. A result of Ward and Zhang suggests that such extension might be possible in the amenable case. In this paper we prove that the addition formula can not be extended for groups of finite type with nonzero Euler-characteristcs e.g. for free groups or surface groups.

math.DS↗
arXiv1998

On quasi-transitive amenable graphs

The existence of nonconstant harmonic Dirichlet functions on a Cayley graph of a discrete group is equivalent to the nonvanishing of the first L2-cohomology of the given group. It was first proven by Cheeger and Gromov that such functions do not exists on the Cayley-graph of an amenable group. The result was extended using Foster's averaging formula to transitive amenable graphs by Medolla and Soardi. In this paper we extend this result further for amenable graphs which are transitive in the category of quasi-isometries. Such graphs are usually not quasi-isometric to transitive graphs. The extension includes nets of amenable unimodular Lie-groups and geometrically amenable homogeneous spaces.

math.GT↗