Endre Szabó

On sofic groups

Abstract Answering some questions of Weiss in [B. Weiss. Sofic groups and dynamical systems. (Ergodic theory and harmonic analysis, Mumbai, 1999.) Sankhya Ser. A. 62 (2000), 350-359.], we prove that a free product of sofic groups is sofic and that amenable extensions of sofic groups are sofic. We also give an example of a finitely generated sofic group that is not residually amenable.

Growth in finite simple groups of Lie type

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^{1+epsilon} where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple groups L of Lie type of bounded rank the diameter of any Cayley graph is polylogarithmic in |L|. We obtain a similar bound for the diameters of all Cayley graphs of perfect subgroups of GL(n,p) generated by their elements of order p. We also obtain some new families of expanders. We also prove the following partial extension. Let G be a subgroup of GL(n,p), p a prime, and S a symmetric set of generators of G satisfying |S^3|\le K|S| for some K. Then G has two normal subgroups H\ge P such that H/P is soluble, P is contained in S^6 and S is covered by K^c cosets of H where c depends on n. We obtain results of similar flavour for sets generating infinite subgroups of GL(n,F), F an arbitrary field.

Sofic representations of amenable groups

Using probabilistic methods, Collins and Dykema proved that the free product of two sofic groups amalgamated over a monotileably amenable subgroup is sofic as well. We show that the restriction is unnecessary; the free product of two sofic groups amalgamated over an arbitrary amenable subgroup is sofic. We also prove a group theoretical analogue of a result of Kenley Jung. A finitely generated group is amenable if and only if it has only one sofic representation up to conjugacy equivalence.

Graphs with automorphism groups admitting composition factors of bounded rank

We prove a 1978 conjecture of Richard Weiss in the case of groups with composition factors of bounded rank. Namely, we prove that there exists a function g : N × N → N such that, for Γ a connected G-vertex-transitive, G-locally primitive graph of valency at most d, if G has no alternating groups of degree greater than r as sections, then a vertex stabiliser in G has size at most g(r, d).

On the product decomposition conjecture for finite simple groups

We prove that if G is a finite simple group of Lie type and S is a subset of G of size at least two, then G is a product of at most c log|G|/log |S| conjugates of S, where c depends only on the Lie rank of G. This confirms a conjecture of Liebeck, Nikolov and Shalev in the case of families of simple groups of bounded rank. We also obtain various related results about products of conjugates of a set within a group.

What has been Wrong with the Retirement Rules in Hungary

A basic problem with the ever-changing Hungarian retirement rules has been that they created excessive shares of gainers and of losers. Certain workers with long (and continuous) employment could retire well below the normal retirement age (NRA) with full benefit. Other workers, with fragmented and therefore short employment had to work until reaching the ever rising NRA. A peculiar consequence of these rules is the strong negative correlation between the retirement age and the length of contribution. Moving in the direction of a fair system, like the Nonfinancial Defined Contribution system, would improve sustainability and fairness.

Nyugdíjba vonulási szabályok Magyarországon - nyertesek és vesztesek

A tanulmany a nyugdijrendszer azon ket szabalyaval foglalkozik, amely meghatarozza, hogyan novekszik a kezdőnyugdij: 1. a befizetett jarulektol es 2. a nyugdijba vonulasi eletkortol fuggően. Vizmertekul az eszmei szamla szolgalhat, amely egy eletbiztositasi- eletjaradekos keny szer meg ta kari tasi rendszert utanoz. Altalanosabban rugalmas korhatarrol beszelunk, ha mind a szolgalati idő, mind a nyugdijba vonulasi kor emelese jelentősen noveli a nyugdijjaradekot, s ezzel osztonoz a nyugdijba vonulas idejenek rugalmas megvalasztasara. Hibas elkepzelesek miatt ez a megkozelites Magyarorszagon csak nagyon torz formaban ervenyesul. Korabban egy engedekeny szabaly szerint az atlag folotti szolgalati idő eseten az előrehozott nyugdijba vonulas nem csokkentette a nyugdijat. Jelenleg ket kiegeszitő szabalyt kell kiemelnunk a) a kivetelező: 2011 ota a Nők40 kereteben minden nő, akinek a jogviszonya eleri a 40 evet, teljes előrehozott nyugdijat kap, b) a merevitő: 2012 ota az a) kategorian kivul senki sem mehet a korhatar alatt nyugdijba - meg csokkentett előrehozott nyugdijjal sem. Bevonva a keresetek es a nyugdij kozotti kapcsolatot megteremtő szabalyt is, a rugalmas esethez kepest ertekeljuk a korabbi es a jelenlegi szabalyok melletti nyerteseket es veszteseket.* Journal of Economic Literature (JEL) kod: H55, I14, J26.