@inbook{10.2307/j.ctt7ztfmd.8, ISBN = {9780691011530}, URL = {http://www.jstor.org/stable/j.ctt7ztfmd.8}, abstract = {Let$\mathcal{T}=\left\langle {{f}_{s}}:{{U}_{s}}\to {{V}_{s}};s\in S \right\rangle$and${{\mathcal{T}}^{\prime }}=\left\langle {{g}_{s}}:U_{s}^{\prime }\to V_{s}^{\prime };s\in S \right\rangle$be a pair of towers with the same level setS. Aconjugacy ϕbetween$\mathcal{T}$and${{\mathcal{T}}^{\prime }}$is a bijection$\phi :\bigcup{{{V}_{s}}}\to \bigcup{V_{s}^{\prime }}$such that$\phi \circ {{f}_{s}}={{g}_{s}}\circ \phi$for all$s\in S$. A conjugacy may be conformal, quasiconformal, smooth, etc. according to the quality ofϕ.A tower$\mathcal{T}$isquasiconformally rigidif any quasiconformal conjugacyϕfrom$\mathcal{T}$to another tower${{\mathcal{T}}^{\prime }}$is conformal.Here are two equivalent formulations of quasiconformal rigidity which make no reference to${{\mathcal{T}}^{\prime }}$. AnL∞Beltrami differential$\mu =\mu (z)d\bar{z}/dz$on$\bigcup {{V}_{s}}$is$\mathcal{T}$-invariantif$f_{s}^{*}(\mu )=\mu |{{U}_{s}}$for all$s\in S$. Then we}, bookauthor = {Curtis T. McMullen}, booktitle = {Renormalization and 3-Manifolds Which Fiber over the Circle (AM-142)}, pages = {105--118}, publisher = {Princeton University Press}, title = {Rigidity of towers}, year = {1996} }