Date:
Thu, 23/11/202310:00-11:00
Link to live broadcast + recording: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=9aa8c5cd-0f3b...
Discrete amenable group is a classic notion that says the following: a given group either admits a left translation invariant probability measure, or has a paradoxical decomposition (see Banach-Tarski paradox). Examples of amenable groups are solvable groups and finitely generated groups of subexponential growth. On the other hand non-abelian free groups and in general non-elementary hyperbolic groups are non-amenable. A notion of amenable algebra was introduced by G. Elek in 2003 and then was extensively developed by L. Bartholdi. It has similarities with the corresponding group definition (for instance, amenable algebra admits a finitely additive invariant dimension-measure). However, unlike groups amenable algebras may have non-amenable subalgebras and quotients.
Now it seems quite interesting to collect examples of amenable and non-amenable algebras for deeper understanding of their structure. In a joint word with A. Kanel-Belov, E. Plotkin and E. Rips we defined small cancellation rings (as an analogue of small cancellation groups and an attempt to use negative curvature for rings) and developed their structure theory. Recently we proved that they are non-amenable. In my talk I will tell what is currently known about amenability of algebras, introduce small cancellation rings and show why they are non-amenable.
Discrete amenable group is a classic notion that says the following: a given group either admits a left translation invariant probability measure, or has a paradoxical decomposition (see Banach-Tarski paradox). Examples of amenable groups are solvable groups and finitely generated groups of subexponential growth. On the other hand non-abelian free groups and in general non-elementary hyperbolic groups are non-amenable. A notion of amenable algebra was introduced by G. Elek in 2003 and then was extensively developed by L. Bartholdi. It has similarities with the corresponding group definition (for instance, amenable algebra admits a finitely additive invariant dimension-measure). However, unlike groups amenable algebras may have non-amenable subalgebras and quotients.
Now it seems quite interesting to collect examples of amenable and non-amenable algebras for deeper understanding of their structure. In a joint word with A. Kanel-Belov, E. Plotkin and E. Rips we defined small cancellation rings (as an analogue of small cancellation groups and an attempt to use negative curvature for rings) and developed their structure theory. Recently we proved that they are non-amenable. In my talk I will tell what is currently known about amenability of algebras, introduce small cancellation rings and show why they are non-amenable.