Finitely summable $\gamma$-elements for word-hyperbolic groups

We present two explicit combinatorial constructions of finitely summable reduced “Gamma”-elements ${\gamma }_r\in KK(C_r^{\ast }(\mathrm{\Gamma }),\mathbb{C})$ for any word-hyperbolic group $(\mathrm{\Gamma },S)$ and obtain summability bounds for them in terms of the cardinality of the generating set $S\subset \mathrm{\Gamma }$ and the hyperbolicity constant of the associated Cayley graph.