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Register a new userThe (2,3)-generation of the special unitary groups of dimension 6
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In this paper we give explicit (2,3)-generators of the unitary groups SU_6(q^ 2), for all q. They fit into a uniform sequence of likely (2,3)-generators for all n>= 6.
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In this paper we prove that the unitary groups $SU_n(q^2)$ are $(2,3)$-generated for any prime power $q$ and any integer $ngeq 8$. By previous results this implies that, if $ngeq 3$, the groups $SU_n(q^2)$ and $PSU_n(q^2)$ are $(2,3)$-generated, except when $(n,q)in{(3,2),(3,3),(3,5),(4,2), (4,3),(5,2)}$.
In this paper we determine the classical simple groups of dimension r=3,5 which are (2,3)-generated (the cases r = 2, 4 are known). If r = 3, they are PSL_3(q), q <> 4, and PSU_3(q^2), q^2 <> 9, 25. If r = 5 they are PSL_5(q), for all q, and PSU_5(q^2), q^2 >= 9. Also, the soluble group PSU_3(4) is not (2,3)-generated. We give explicit (2,3)-generators of the linear preimages, in the special linear groups, of the (2,3)-generated simple groups.
This paper is a new important step towards the complete classification of the finite simple groups which are $(2, 3)$-generated. In fact, we prove that the symplectic groups $Sp_{2n}(q)$ are $(2,3)$-generated for all $ngeq 4$. Because of the existing literature, this result implies that the groups $PSp_{2n}(q)$ are $(2,3)$-generated for all $ngeq 2$, with the exception of $PSp_4(2^f)$ and $PSp_4(3^f)$.
For each $n$ we construct examples of finitely presented $C(1/6)$ small cancellation groups that do not act properly on any $n$-dimensional CAT(0) cube complex.
Let $G$ be a virtually special group. Then the residual finiteness growth of $G$ is at most linear. This result cannot be found by embedding $G$ into a special linear group. Indeed, the special linear group $text{SL}_k(mathbb{Z})$, for $k > 2$, has residual finiteness growth $n^{k-1}$.
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