ACC Seminar: On the Dual of Commensurability and Virtual Embeddings into Direct Products

ABSTRACT

Groups  G_1  and  G_2  are commensurable if there is a third group  H  that embeds as a finite index subgroup in  G_1  and  G_2.  We can "reverse the arrows" and declare  G_1,  G_2  to be co-commensurable if they both embed as finite index subgroups of a common overgroup. Co-commensurability implies commensurability.  An exploration of the methods to construct common finite covers of graphs led me to an exploration of when a commensuration can be induced by a co-commensuration. In this presentation I will cover the motivating problem as well as the solution to the problem of when a commensuration can be completed to a co-commensuration. I will then discuss the unexpected contribution of Ashot Minasyan who saw that my ideas had implications towards the virtual embeddings of groups into direct products. This topic turns out to be related to the answer to a question about Brough's conjecture that Paul Schupp asked when he was visiting Stevens and tangentially related to recent work of Shen and Ushakov.

BIOGRAPHY

Nicholas Touikan is Associate Professor at the Department of Mathematics and Statistics of the University of New Brunswick. Professor Touikan received his PhD degree from McGill University under the direction of Olga Kharlampovich. His research interests include algorithmic problems for non-positively curved groups, and connections between low dimensional topology and group actions.

Attendance: This is a technical talk open to all.
A campus map is available at https://tour.stevens.edu.
Additional information is available at https://web.stevens.edu/algebraic/.