arXiv subjects
Publications and source records attributed to Martin Kassabov.
We construct an explicit generating sets $F_n$ and $\tilde F_n$ of the alternating and the symmetric groups, which make the Cayley graphs $C(Alt(n), F_n)$ and $C(Sym(n), \tilde F_n)$ 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.
We prove that the universal lattices -- the groups $G=\SL_d(R)$ where $R=\Z[x_1,...,x_k]$, have property $τ$ for $d\geq 3$. 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 expanding 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 $K_2$ of some finite commutative rings and a relative property \emph{T} of $(\SL_2(R) \ltimes R^2, R^2)$.
We study the representations of non-commutative universal lattices and use them to compute lower bounds for the \TauC for the commutative universal lattices $G_{d,k}= \SL_d(\Z[x_1,...,x_k])$ with respect to several generating sets. As an application of the above result we show that the Cayley graphs of the finite groups $\SL_{3k}(\F_p)$ can be made expanders using suitable choice of the generators. This provides the first examples of expander families of groups of Lie type where the rank is not bounded and gives a natural (and explicit) counter examples to two conjectures of Alex Lubotzky and Benjamin Weiss.
In this article we introduce the notion of weak identities in a group and study their properties. We show that weak identities have some similar properties to ordinary ones. We use this notion to prove that any finitely generated solvable discriminating group is abelian.
In this article we improve the known Kazhdan constant for $SL_n(Z)$ with respect to the generating set of the elementary matrices. We prove that the Kazhdan constant is bounded from below by $[42\sqrt{n}+860]^{-1}$, which gives the exact asymptotic behavior of the Kazhdan constant, as $n$ goes to infinity, since $\sqrt{2/n}$ is an upper bound. We can use this bound to improve the bounds for the spectral gap of the Cayley graph of $SL_n(F_p)$ and for the working time of the product replacement algorithm for abelian groups.
In this thesis I study the automorphism tower of free nilpotent groups. Our main tool in studying the automorphism tower is to embed every group as a lattice in some Lie group. Using known rigidity results the automorphism group of the discrete group can be embedded into the automorphism group of the Lie group. This allows me to lift the description of the derivation tower of the free nilpotent Lie algebra to obtain information about the automorphism tower of the free nilpotent group. The main result in this thesis states that the automorphism tower of the free nilpotent group $Γ(n,d)$ on $n$ generators and nilpotency class $d$, stabilizes after finitely many steps. If the nilpotency class is small compared to the number of generators we have that the height of the automorphism tower is at most 3.
We construct an example of an IA-automorphism of the free metabelian group of rank $n\geq 3$ without nontrivial fixed points. That gives a negative answer to the question raised by Shpilrain. By a result of Bachmuth \cite{Ba1}, such an automorphism does not exist if the rank is equal to 2.
Let $Σ_{g,b}$ denote a closed oriented surface genus $g$ with $b$ punctures and let $Mod_{g,b}$ denote its mapping class group. Luo proved that if the genus is at least 3, the group $Mod_{g,b}$ is generated by involutions. He also asked if there exists a universal upper bound, independent of genus and the number of punctures, for the number of torsion elements/involutions needed to generate $Mod_{g,b}$. Brendle and Farb gave a partial answer in the case of closed surfaces and surfaces with one puncture, by describing a generating set consisting of 7 involutions. Our main result generalizes the above result to the case of multiple punctures. We also show that the mapping class group can be generated by smaller number of involutions. More precisely, we prove that the mapping class group can be generated by 4 involutions if the genus $g$ is large enough. There is not a lot room to improve this bound because to generate this group we need at lest 3 involutions. In the case of small genus (but at least 3) to generate the whole mapping class group we need a few more involutions.