Adding high powered relations to large groups

Abstract

It is a common theme in group theory that if a group G is quotiented by a sufficiently high power of an element g ∈ G, then properties of G are inherited by the quotient group. A major example of this phenomenon is the theorem of Gromov ([4], see also [2]) asserting that if G is torsion-free and word-hyperbolic, then G/〈〈g〉〉 is also word-hyperbolic for all sufficiently big integers n. In this paper, we will prove that this phenomenon occurs for the property of being large. Recall that G is large if some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. Large groups have many interesting and useful properties, including super-exponential subgroup growth and infinite virtual first Betti number. They are also particularly important in low-dimensional topology. Our main theorem is the following.