### A combinatorial higher rank hyperbolicity condition

Analysis and Geometry Seminar

3rd February 2022, 3:15 pm – 4:15 pm

Online, Contact organisers for details

The talk reports on joint work with Martina Jorgensen, which is based in turn on joint works with Bruce Kleiner and with Tommaso Goldhirsch.We introduce and study a coarse version of a *2(n+1)*-point inequality characterizing metric spaces of combinatorial dimension at most *n* due to Dress. This condition, experimentally called *(n,δ)*-hyperbolicity, reduces to Gromov's quadruple definition of *δ*-hyperbolicity in case *n = 1*. The *ℓ _{∞}*-product of

*n*

*δ*-hyperbolic spaces is

*(n,δ)*-hyperbolic. We show that every

*(n,δ)*-hyperbolic space possesses a slim

*(n+1)*-simplex property. We relate this condition to other instances of rank-

*n*-hyperbolicity and verify it for certain classes of spaces and groups.

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