### Qiang Zeng (University of Illinois at Urbana-Champaign)

Title: Subgaussian 1-cocycles on discrete groups

Abstract: We prove the $L_p$ Poincar\'e inequalities with constant $C\sqrt{p}$ for $1$-cocycles on countable discrete groups under Bakry--Emery's $\Ga_2$-criterion. These inequalities determine an analogue of subgaussian behavior for 1-cocycles. Our theorem in particular implies Efraim and Lust-Piquard's Poincar\'e type inequalities for the Walsh system. As complementary results, we also show that in the noncommutative diffusion setting the spectral gap inequality implies the $L_p$ Poincar\'e inequalities with constant $C{p}$ under some conditions. Joint work with Marius Junge.