Martin Kassabov

Finite simple groups as expanders

We prove that there exist k in and 0 < epsilon in such that every non-abelian finite simple group G, which is not a Suzuki group, has a set of k generators for which the Cayley graph Cay(G; S) is an epsilon-expander.

KAZHDAN CONSTANTS FOR SLn(ℤ)

In this article we improve the known Kazhdan constant for SLn(ℤ) with respect to the generating set of the elementary matrices. We prove that the Kazhdan constant is bounded from below by , which gives the exact asymptotic behavior of the Kazhdan constant, as n goes to infinity, since is an upper bound. We can use this bound to improve the bounds for the spectral gap of the Cayley graph of SLn(𝔽p) and for the working time of the product replacement algorithm for abelian groups.

Hairy graphs and the unstable homology of Mod(g, s), Out(Fn) and Aut(Fn)

We study a family of Lie algebras {hO} which are defined for cyclic operads O. Using his graph homology theory, Kontsevich identified the homology of two of these Lie algebras (corresponding to the Lie and associative operads) with the cohomology of outer automorphism groups of free groups and mapping class groups of punctured surfaces, respectively. In this paper we introduce a hairy graph homology theory for O. We show that the homology of hO embeds in hairy graph homology via a trace map which generalizes the trace map defined by S. Morita. For the Lie operad we use the trace map to find large new summands of the abelianization of hO which are related to classical modular forms for SL(2,Z). Using cusp forms we construct new cycles for the unstable homology of Out(F_n), and using Eisenstein series we find new cycles for Aut(F_n). For the associative operad we compute the first homology of the hairy graph complex by adapting an argument of Morita, Sakasai and Suzuki, who determined the complete abelianization of hO in the associative case.

Presentations of finite simple groups: A quantitative approach

There is a constant such that all nonabelian finite simple groups of rank over , with the possible exception of the Ree groups , have presentations with at most generators and relations and total length at most . As a corollary, we deduce a conjecture of Holt: there is a constant such that for every finite simple group , every prime and every irreducible -module

BOUNDING THE RESIDUAL FINITENESS OF FREE GROUPS

We find a lower bound to the size of finite groups detecting a given word in the free group, more precisely we construct a word wn of length n in non-abelian free groups with the property that wn is the identity on all finite quotients of size ? n2/3 or less. This improves on a previous result of Bou- Rabee and McReynolds quantifying the lower bound of the residual finiteness of free groups.

Presentations of finite simple groups: profinite and cohomological approaches

We prove the following three closely related results. The first is that every finite simple group has a profinite presentation with 2 generators and at most 18 relations. The second is that if G is a finite simple group, F a field and M an FG-module, then the dimension of the second cohomology group of G with coefficients in M is at most 17.5 times the dimension of M. The third result is that we may replace 17.5 by 18.5 as long as M is faithful irreducible G-module. These last two results answer conjectures of Holt.

Universal lattices and property tau

We prove that the universal lattices – the groups G=SLd(R) where R=ℤ[x1,...,xk], have property τ for d≥3. This provides the first example of linear groups with τ which do not come from arithmetic groups. We also give a lower bound for the τ-constant with respect to the natural generating set of G. Our methods are based on bounded elementary generation of the finite congruence images of G, a generalization of a result by Dennis and Stein on K2 of some finite commutative rings and a relative property T of

Symmetric groups and expanders

(\mathrm{SL}_2(R)\ltimes R^2, R^2)

Presentations of finite simple groups: a computational approach

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Cartesian products as profinite completions

We construct an explicit generating sets Fn and ˜ Fn of the alternating and the symmetric groups, which make the Cayley graphs C(Alt(n), Fn) and C(Sym(n), ˜ Fn) a family of bounded degree expanders for all sufficiently large n. These expanders have many applications in the theory of random walks on groups and other areas of mathematics. A finite graph is called an ǫ-expander for some ǫ ∈ (0,1), if for any subset A ⊆ of size at most | |/2 we have |∂(A)| > ǫ|A| (where ∂(A) is the set of vertices of \A of edge distance 1 to A). The largest such ǫ is called the expanding constant of . Constructing families of ǫ-expanders with bounded valency is an important practical problem in computer science, because such graphs have many nice properties — for example they have a logarithmic diameter. For an excellent introduction to the subject we refer the reader to the book [14] by A. Lubotzky. Using counting arguments it can be shown that almost any 5 regular graph is 1/5-expander. However constructing an explicit examples of families expander graphs is a difficult problem. The first explicit construction of a family of expanders was done by G. Margulis in [19], using Kazhdan property T of SL3(Z). Currently there are several different construction of expanders. With the exception of a few recent ones based on the zig-zag products of graphs (see [2, 23, 24]), all constructions are based groups theory and use some variant of property T (property τ, Selberg property etc.). Kazhdan Property T is not very interesting for a given finite group G (all finite groups have property T), but the related Kazhdan constant with respect to some generating F set is. Given an infinite collection of finite groups Gi, it is a challenge to prove the existence of uniform Kazhdan constants with respect to properly chosen generating sets. This problem is related to construction a family of expanders using the Cayley graphs of the groups Gi. The original definition of property T uses the Fell topology of the unitary dual, see [11]. Here we will use an equivalent definition (only for discrete groups) which also addresses the notion of the Kazhdan constants.