Let $/Gamma$ be a nonamenable finitely generated infinite hyperbolic group with a symmetric generating set $S,$ and $/partial/Gamma$ the hyperbolic boundary of its Cayley graph. Fix a symmetric probability $/mu$ on $/Gamma$ whose support is $S,$ and denote by $/rho=/rho(/mu)$ the spectral radius of the random walk $/xi$ on $/Gamma$ associated to $/mu.$ Let $/nu$ be a probability on $/{1,2,3,/cdots/}$ with a finite mean $/lambda.$ Write $/Lambda/subseteq/partial/Gamma$ for the boundary of the branching random walk with offspring distribution $/nu$ and underlying random walk $/xi,$ and $h(/nu)$ for the Hausdorff dimension of $/Lambda.$ When $/lambda>1//rho,$ the branching random walk is recurrent, trivially$$/Lambda=/partial/Gamma,/ h(/nu)=/dim(/partial /Gamma).$$In this talk, we focus on the transient setting i.e. $/lambda/in [1,1//rho],$ and prove the following results: $h(/nu)$ is a deterministic function of $/lambda$ and thus denote it by $h(/lambda);$ and $h(/lambda)$ is continuous and strictly increasing in $/lambda/in [1,1//rho]$ and $h(1//rho)/leq/frac{1}{2}/dim(/partial /Gamma);$ and there is a positive constant $C$ such that$$h(1//rho)-h(/lambda)/sim C/sqrt{1//rho-/lambda}/ /mbox{as}/ /lambda/uparrow 1//rho.$$The above results confirm a conjecture of S. Lalley in his ICM 2006 Lecture (the critical exponent of Hausdorff dimensions of boundaries of branching random walks on hyperbolic groups is $1/2$).This talk is based on a joint work with Shi Zhan, Sidoravicius Vladas and Wang Longmin.

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