#### IMSc Webinar

#### Asymptotics of powers in finite reductive groups

#### Anupam Kumar Singh

##### IISER Pune

*Zoom link: us02web.zoom.us/j/88975543567*

Let G be a connected reductive group defined over a finite field F_q. Fix an

integer M >1, and consider the power map x going to x^M on G. We denote the image of G(F_q) under this map by G(F_q)^M and estimate what proportion of regular semisimple, semisimple and regular elements of G(F_q) it contains. We prove that as q tends to infinity, all of these proportions are equal and provide a formula for the same. We also calculate this more explicitly for the groups GL(n, q) and U(n, q).

Done