Existence of finitely presented intersection-saturated groups

(This is joint work with J. Delgado and M. Roy) For two subgroups of a group, $H_1, H_2\leq G$, we can look at the eight possibilities for the finite/non-finite generability of $H_1$, $H_2$, and $H_1\cap H_2$. For example, all eight are possible in a free non-abelian group except one of them, expressing the well-known fact that free groups are Howson: intersection of two finitely generated subgroups is again finitely generated. A group $G$ is called intersection-saturated when, for every $k$, each of the $2^{2^k-1}$ such $k$-configurations are realizable by appropriate subgroups $H_1,\ldots ,H_k\leq G$. 


In this talk we construct explicit finitely presented intersection-saturated groups. We also show that the Howson property is the only restriction for realizability in free groups: a $k$-configuration is realizable in a free non-abelian group if and only if it respects the Howson property. 


If time permits I will explain some ideas to dualize the situation and be able to realize quotient $k$-configurations (this is still work in progress by the same authors).

Date and Venue

Start Date
FC1 029
End Date


Enric Ventura

Speaker's Institution

Universitat Politècnica de Catalunya



Algebra, Combinatorics and Number Theory