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We prove that all finite simple groups of Lie type, with the exception of the Suzuki groups, can be made into a family of expanders in a uniform way. This confirms a conjecture of Babai, Kantor and Lubotzky from 1989, which has already been proved by Kassabov for sufficiently large rank. The bounded rank case is deduced here from a uniform result for ${SL}_2$ which is obtained by combining results of Selberg and Drinfeld via an explicit construction of Ramanujan graphs by Lubotzky, Samuels and Vishne.

Let $\Gamma$ be a group and $r_n\left(\Gamma \right)$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function ${𝒵}_{\Gamma }\left(s\right)={\sum }_{n=1}^{\infty }{r}_n\left(\Gamma \right)n^{-s}$. When $\Gamma$ is an arithmetic group satisfying the congruence subgroup property then ${𝒵}_{\Gamma }\left(s\right)$ has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place...

Let $2\le a\le b\le c\in \mathbb{N}$ with $\mu =1/a+1/b+1/c<1$ and let $T=T_{a,b,c}=\langle x,y,z:x^a=y^b=z^c=xyz=1\rangle$ be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of $T$? (Classically, for $(a,b,c)=(2,3,7)$ and more recently also for general $(a,b,c)$.) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially prove...

All finite simple groups of Lie type of rank $n$ over a field of size $q$, with the possible exception of the Ree groups ${}^2{G}_2\left(q\right)$, have presentations with at most 49 relations and bit-length $O(\mathrm{𝚕𝚘𝚐}n+\mathrm{𝚕𝚘𝚐}q)$. Moreover, $A_n$ and $S_n$ have presentations with 3 generators; 7 relations and bit-length $O(\mathrm{𝚕𝚘𝚐}n)$, while $\mathrm{𝚂𝙻}(n,q)$ has a presentation with 6 generators, 25 relations and bit-length $O(\mathrm{𝚕𝚘𝚐}n+\mathrm{𝚕𝚘𝚐}q)$.