Date:
Tue, 14/12/202114:00-15:00
Abstract:
The space Hom(\Gamma,G) of homomorphisms from a finitely-generated group \Gamma to a complex semisimple algebraic group G is known as the G-representation variety of \Gamma. We study this space when G is fixed and \Gamma is a random group in the few-relators model. That is, \Gamma is generated by k elements subject to r random relations of length L, where k and r are fixed and L tends to infinity.
More precisely, we study the subvariety Z of Hom(\Gamma,G), consisting of all homomorphisms whose images are Zariski dense in G. We give an explicit formula for the dimension of Z, valid with probability tending to 1, and study the Galois action on its geometric components. In particular, we show that in the case of deficiency 1 (i.e., k-r=1), the Zariski-dense G-representations of a typical \Gamma enjoy Galois rigidity.
Our methods assume the Generalized Riemann Hypothesis and exploit mixing of random walks and spectral gap estimates on finite groups.
Based on a joint work with E. Breuillard and P. Varju.
Zoom details:
Join Zoom Meeting
Meeting ID: 881 3835 5697
Passcode: 904821
הצטרפות באמצעות Google Meet: https://meet.google.com/qub-nmaf-pud
למידע נוסף על Meet, אפשר להיכנס לאתר: https://support.google.com/a/users/answer/9282720
The space Hom(\Gamma,G) of homomorphisms from a finitely-generated group \Gamma to a complex semisimple algebraic group G is known as the G-representation variety of \Gamma. We study this space when G is fixed and \Gamma is a random group in the few-relators model. That is, \Gamma is generated by k elements subject to r random relations of length L, where k and r are fixed and L tends to infinity.
More precisely, we study the subvariety Z of Hom(\Gamma,G), consisting of all homomorphisms whose images are Zariski dense in G. We give an explicit formula for the dimension of Z, valid with probability tending to 1, and study the Galois action on its geometric components. In particular, we show that in the case of deficiency 1 (i.e., k-r=1), the Zariski-dense G-representations of a typical \Gamma enjoy Galois rigidity.
Our methods assume the Generalized Riemann Hypothesis and exploit mixing of random walks and spectral gap estimates on finite groups.
Based on a joint work with E. Breuillard and P. Varju.
Zoom details:
Join Zoom Meeting
Meeting ID: 881 3835 5697
Passcode: 904821
הצטרפות באמצעות Google Meet: https://meet.google.com/qub-nmaf-pud
למידע נוסף על Meet, אפשר להיכנס לאתר: https://support.google.com/a/users/answer/9282720