Descripción del proyecto
Una investigación aprovecha los subgrupos aleatorios estacionarios para estudiar variedades
Los subgrupos aleatorios invariantes han demostrado ser muy útiles para estudiar redes y sus invariantes asintóticas. Con todo, la restricción a medidas invariantes limita el número de problemas que se pueden investigar. El proyecto SRS, financiado con fondos europeos, aprovechará los subgrupos aleatorios estacionarios, que son mucho más generales, y ayudará a analizar subgrupos discretos de grupos de covolumen infinito, sobre todo de subgrupos finos de grupos aritméticos. Sus investigadores tratarán de resolver la variante de la conjetura de Schoen-Yau postulada por Margulis, es decir, las variedades localmente simétricas de rango superior Λ\G/K de volumen infinito que no son variantes de Liouville. Una respuesta positiva tendría muchas aplicaciones en la teoría de subgrupos discretos de grupos de Lie.
Objetivo
The notion of invariant random subgroups (IRS) has proven extremely useful during the last decade, particularly to the study of asymptotic invariants of lattices. However, the scope of problems that one can investigate when restricting to invariant measures (on the space of subgroups) is limited. It was recently realised that the notion of stationary random subgroups (SRS), which is much more general, is still extremely powerful and opens up new paths to attacking problems that previously seemed to be out of our reach.
Notably, the notion of stationary random subgroups has turned out to be a wonderful new tool in the analysis of discrete subgroups of infinite co-volume, and, in particular, thin subgroups of arithmetic groups. A few months ago M. Fraczyk and I proved, using SRS, the following conjecture of Margulis: Let G be a higher rank simple Lie group and Λ ⊂ G a discrete subgroup. Then the orbifold Λ\G/K has finite volume if and only if it has bounded injectivity radius. This is a far-reaching generalisation of the celebrated Normal Subgroup Theorem of Margulis, and while it is new even for subgroups of lattices, it is completely general.
One of the main problems we wish to solve is the variant of the Schoen–Yau Conjecture postulated by Margulis; namely, that higher rank, locally symmetric manifolds Λ\G/K of infinite volume are not Liouville. A positive answer would have many applications in the theory of discrete subgroups of Lie groups. Some exiting applications are possible using partial results.
Palabras clave
Programa(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Régimen de financiación
HORIZON-ERC - HORIZON ERC GrantsInstitución de acogida
7610001 Rehovot
Israel