Realizable ranks of joins and intersections of subgroups in free groups

Ignat Soroko

Published 2019-01-14Version 1

The famous Hanna Neumann Theorem gives an upper bound for the ranks of the intersection of arbitrary subgroups $H$ and $K$ of a non-abelian free group. It is an interesting question to "quantify" this bound with respect to the rank of $H\vee K$, the subgroup generated by $H$ and $K$. We describe a set of realizable values $(rk(H\vee K),rk(H\cap K))$ for arbitrary $H$, $K$, and conjecture that this locus is complete. We study the combinatorial structure of Kent's topological pushout of the core graphs for $H$ and $K$, with the help of graphs introduced by Dicks in the context of his Amalgamated Graph Conjecture. Using it, we show that the conditions $rk(H\vee K)=rk(H)+rk(K)-i$, $rk(H\cap K)=\frac{i(i-1)}2+1$, for $i\ge 3$ are not realizable, thus resolving the remaining open case $m=4$ of Guzman's "Group-Theoretic Conjecture" in the affirmative, which in turn implies the validity of the corresponding "Geometric Conjecture" on hyperbolic $3$-manifolds with a $6$-free fundamental group.