We give a description of non-growing subsets in linear groups, which extends the Product theorem for simple groups of Lie type. We also give an account of various related aspects of growth in linear groups.

Around twenty years ago Ghys conjectured that finite subgroups of the diffeomorphism group of a compact smooth manifold M have an abelian normal subgroup of index at most a(M), where a(M) depends only on M. First we construct a family of counterexamples to this conjecture including, for example, the product space T^2× S^2. Following the first appearance of our counterexample on the arXiv Ghys put forward a revised conjecture, which predicts only the existence of a nilpotent normal subgroup of index at most n(M). Our main result is the proof of the revised Ghys conjecture. More generally, we show that the same result holds for homeomorphism groups of not necessarily compact topological manifolds with finitely generated homology groups. Our proofs are based on finite group theoretic results which provide a general strategy for proving similar Jordan-type theorems.

Let $$G$$ be a non-abelian finite simple group. A famous resultof Liebeck and Shalev is that there is an absolute constant $$c$$ such that whenever $$S$$ is a non-trivial normal subset in $$G$$ then $$S^{k} = G$$ for any integer $$k$$ at least $$c \cdot (\log|G|/\log|S|)$$ . This result is generalized by showing that there exists an absoluteconstant $$c$$ such that whenever $$S_{1}$$ , $$\ldots , $$ $$S_{k}$$ are normal subsets in $$G$$ with $$\prod_{i=1}^{k} |S_{i}| \geq {|G|}^{c}$$ then $$S_{1} \cdots S_{k} = G$$ .

Noether, Fleischmann and Fogarty proved that if the characteristic of the underlying field does not divide the order $|G|$ of a finite group $G$, then the polynomial invariants of $G$ are generated by polynomials of degrees at most $|G|$. Let $\beta(G)$ denote the largest indispensable degree in such generating sets. Cziszter and Domokos recently described finite groups $G$ with $|G|/\beta(G)$ at most $2$. We prove an asymptotic extension of their result. Namely, $|G|/\beta(G)$ is bounded for a finite group $G$ if and only if $G$ has a characteristic cyclic subgroup of bounded index. In the course of the proof we obtain the following surprising result. If $S$ is a finite simple group of Lie type or a sporadic group then we have $\beta(S) \leq {|S|}^{39/40}$. We ask a number of questions motivated by our results.

For a finite group G, let diam (G) denote the maximum diameter of a connected Cayley graph of G. A well-known conjecture of Babai states that diam (G) is bounded by (log2|G|)O(1) in case G is a non-abelian finite simple group. Let G be a finite simple group of Lie type of Lie rank n over the field Fq. Babai's conjecture has been verified in case n is bounded, but it is wide open in case n is unbounded. Recently, Biswas and Yang proved that diam (G) is bounded by qO(n(log2n+log2q)3). We show that in fact diam (G)<qO(n(log2n)2) holds. Note that our bound is significantly smaller than the order of G for n large, even if q is large. As an application, we show that more generally diam (H)<qO(n(log2n)2) holds for any subgroup H of GL (V), where V is a vector space of dimension n defined over the field Fq.

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 the diameter of any Cayley graph is polylogarithmic in |L|. Combining our result on growth with known results of Bourgain,Gamburd and Varj\'u it follows that if LAMBDA is a Zariski-dense subgroup of SL(d,Z) generated by a finite symmetric set S, then for square-free moduli m which are relatively prime to some number m_0 the Cayley graphs Gamma(SL(d,m),pi_m(S)) form an expander family.

We obtain a weaker but still quasipolynomial version of Babai's famous GI-algorithm.

We show that if $M$ is a compact smooth manifold diffeomorphic to the total space of an orientable $S^2$ bundle over the torus $T^2$, then its diffeomorphism group does not have the Jordan property, i.e., Diff$(M)$ contains a finite subgroup $G_n$ for any natural number $n$ such that every abelian subgroup of $G_n$ has index at leat $n$. This gives a counterexample to an old conjecture of Ghys.

We prove that every connected strongly regular graph on sufficiently many vertices is Hamiltonian. We prove this by showing that, apart from three families, connected strongly regular graphs are (highly) pseudo-random. Our results suggest a number of new questions and conjectures.

A palyazat resztvevői igen aktivak voltak a 2006-2008 evekben. Nemcsak sok eredmenyt ertek el, miket tobb mint 150 cikkben publikaltak, eredmenyesen nepszerűsitettek azokat. Tobb mint 100 konferencian vettek reszt es adtak elő, felereszben meghivott, vagy plenaris előadokent. Hagyomanyos grafelmelet Tobb extremalis grafproblemat oldottunk meg. Uj eredmenyeket kaptunk Ramsey szamokrol, globalis es lokalis kromatikus szamokrol, Hamiltonkorok letezeseseről. a crossig numberről, graf kapacitasokrol es kizart reszgrafokrol. Veletlen grafok, nagy grafok, regularitasi lemma Nagy grafok hasonlosagait vizsgaltuk. Kulonfele metrikak ekvivalensek. Űj eredemenyeink: Hereditary Property Testing, Inverse Counting Lemma and the Uniqueness of Hypergraph Limit. Hipergrafok, egyeb kombinatorika Uj Sperner tipusu tetelekte kaptunk, aszimptotikusan meghatarozva a halmazok max szamat bizonyos kizart struktőrak eseten. Tobb esetre megoldottuk a kizart hipergraf problemat is. Elmeleti szamitastudomany Uj ujjlenyomat kodokat es bioinformatikai eredmenyeket kaptunk. | The participants of the project were scientifically very active during the years 2006-2008. They did not only obtain many results, which are contained in their more than 150 papers appeared in strong journals, but effectively disseminated them in the scientific community. They participated and gave lectures in more than 100 conferences (with multiplicity), half of them were plenary or invited talks. Traditional graph theory Several extremal problems for graphs were solved. We obtained new results for certain Ramsey numbers, (local and global) chromatic numbers, existence of Hamiltonian cycles crossing numbers, graph capacities, and excluded subgraphs. Random graphs, large graphs, regularity lemma The similarities of large graphs were studied. We show that several different definitions of the metrics (and convergence) are equivalent. Several new results like the Hereditary Property Testing, Inverse Counting Lemma and the Uniqueness of Hypergraph Limit were proved Hypergraphs, other combinatorics New Sperner type theorems were obtained, asymptotically determining the maximum number of sets in a family of subsets with certain excluded configurations. Several cases of the excluded hypergraph problem were solved. Theoretical computer science New fingerprint codes and results in bioinformatics were found.

We prove that if G is a finite almost simple group, having socle of Lie type of rank r, then the number of maximal subgroups of G is at most Cr−2/3|G|, where C is an absolute constant. This verifies a conjecture of Wall for groups of sufficiently large rank. Using this we prove that any finite group G has at most 2C|G|3/2 maximal subgroups.

Let w be a non-trivial word in two variables. We prove that the probability that two randomly chosen elements x, y of a nonabelian finite simple group S satisfy w(x, y) = 1 tends to 0 as |S| → ∞. As a consequence, we obtain a new short proof of a well-known conjecture of Magnus concerning free groups, as well as some applications to profinite groups. Research partially supported by NSERC grant A7171 for J.D., the Hungarian Academy of Sciences grant AKP 96/2-675 for L.P., NSF grants CCR-9503430, CCR-9731799 for Á. S. and a grant from the Israel Science Foundation for A. S. 1991 Mathematics Subject Classification: 20D06, 20E05, 20E26, 20P05.

The author proves in this paper that every profinite group G with polynomial subgroup growth is boundedly generated; that is, it is a product of finitely many procyclic subgroups. This answers a question of P. Zalesskii. By contrast, if G is a boundedly generated group, then the subgroup growth of G is at most nclogn . As a byproduct, a short, elementary proof demonstrates that Aut(Fr) (for r [ges ] 2) and many other related groups are not boundedly generated.

We extend a result of E. Hrushovski and A. Pillay as follows. Let G be a finite subgroup of GL(n,F) where F is a field of characteristic p such that p is sufficiently large compared to n. Assume thatG is generated by p-elements. ThenG is a product of 25 of its Sylow p-subgroups. If G is a simple group of Lie type in characteristic p, the analogous result holds without any restriction on the Lie rank of G. We also give an application of the Hrushovski-Pillay result showing that finitely generated adelic profinite groups are boundedly generated (i.e., such a group is a product of finitely many closed procyclic subgroups). This confirms a conjecture of V. Platonov and B. Sury which was motivated by characterizations of the congruence subgroup property for arithmetic groups.

We prove that if the set of commuting pairs of a profinite group G has positive Haar measure then G is abelian by finite. Using this we show that the set I of involutions has positive measure exactly if I contains a nonempty open subset of G.

vertices has diameter at most 5 logn. This essentially settles a problem of Brouwer, Cohen and Neumaier.

Let A be a group of automorphisms of the finite group G such that (∣A∣, ∣G∣)=1. Then ∣A∣<∣G∣2, and the exponent 2 here is best possible. If, moreover, A is nilpotent of class at most 2, then ∣A∣<∣G∣. If A is abelian, then A has a regular orbit on G. 1991 Mathematics Subject Classification 20D45.