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Register a new userFactorizations of almost simple orthogonal groups of minus type
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This is the fourth one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with orthogonal groups of minus type.
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For a finite group $G$, let $mathrm{diam}(G)$ denote the maximum diameter of a connected Cayley graph of $G$. A well-known conjecture of Babai states that $mathrm{diam}(G)$ is bounded by ${(log_{2} |G|)}^{O(1)}$ 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 $F_{q}$. Babais 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 $mathrm{diam}(G)$ is bounded by $q^{O( n {(log_{2}n + log_{2}q)}^{3})}$. We show that in fact $mathrm{diam}(G) < q^{O(n {(log_{2}n)}^{2})}$ 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 $mathrm{diam}(H) < q^{O( n {(log_{2}n)}^{2})}$ holds for any subgroup $H$ of $mathrm{GL}(V)$, where $V$ is a vector space of dimension $n$ defined over the field $F_q$.
Recursive algebraic construction of two infinite families of polynomials in $n$ variables is proposed as a uniform method applicable to every semisimple Lie group of rank $n$. Its result recognizes Chebyshev polynomials of the first and second kind as the special case of the simple group of type $A_1$. The obtained not Laurent-type polynomials are proved to be equivalent to the partial cases of the Macdonald symmetric polynomials. Basic relation between the polynomials and their properties follow from the corresponding properties of the orbit functions, namely the orthogonality and discretization. Recurrence relations are shown for the Lie groups of types $A_1$, $A_2$, $A_3$, $C_2$, $C_3$, $G_2$, and $B_3$ together with lowest polynomials.
Let $G$ be a simple algebraic group over an algebraically closed field and let $X$ be an irreducible subvariety of $G^r$ with $r geqslant 2$. In this paper, we consider the general problem of determining if there exists a tuple $(x_1, ldots, x_r) in X$ such that $langle x_1, ldots, x_r rangle$ is Zariski dense in $G$. We are primarily interested in the case where $X = C_1 times cdots times C_r$ and each $C_i$ is a conjugacy class of $G$ comprising elements of prime order modulo the center of $G$. In this setting, our main theorem gives a complete solution to the problem when $G$ is a symplectic or orthogonal group. By combining our results with earlier work on linear and exceptional groups, this gives a complete solution for all simple algebraic groups. We also present several applications. For example, we use our main theorem to show that many faithful representations of symplectic and orthogonal groups are generically free. We also establish new asymptotic results on the probabilistic generation of finite simple groups by pairs of prime order elements, completing a line of research initiated by Liebeck and Shalev over 25 years ago.
Let $G$ be a finite simple group of Lie type, and let $pi_G$ be the permutation representation of $G$ associated with the action of $G$ on itself by conjugation. We prove that every irreducible representation of $G$ is a constituent of $pi_G$, unless $G=PSU_n(q)$ and $n$ is coprime to $2(q+1)$, where precisely one irreducible representation fails. Let St be the Steinberg representation of $G$. We prove that a complex irreducible representation of $G$ is a constituent of the tensor square $Stotimes St$, with the same exceptions as in the previous statement.
We show that the group of almost automorphisms of a d-regular tree does not admit lattices. As far as we know this is the first such example among (compactly generated) simple locally compact groups.
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