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In this paper we study the (2,k)-generation of the finite classical groups SL(4,q), Sp(4,q), SU(4,q^2) and their projective images. Here k is the order of an arbitrary element of SL(2,q), subject to the necessary condition k>= 3. When q is even we allow also k=4.
We give a necessary and sufficient condition for a 2-dimensional or a three-generator Artin group $A$ to be (virtually) cocompactly cubulated, in terms of the defining graph of $A$.
We show that many 2-dimensional Artin groups are residually finite. This includes 3-generator Artin groups with labels $geq$ 3 where either at least one label is even, or at most one label is equal 3. As a first step towards residual finiteness we sh
A group G is called bounded if every conjugation-invariant norm on G has finite diameter. We introduce various strengthenings of this property and investigate them in several classes of groups including semisimple Lie groups, arithmetic groups and li
Previously, the authors proved that the presentation complex of a one-relator group $G$ satisfies a geometric condition called negative immersions if every two-generator, one-relator subgroup of $G$ is free. Here, we prove that one-relator groups wit
A subset $S$ of a group $G$ invariably generates $G$ if $G= langle s^{g(s)} | s in Srangle$ for every choice of $g(s) in G,s in S$. We say that a group $G$ is invariably generated if such $S$ exists, or equivalently if $S=G$ invariably generates $G$.