Amenable groups with nearly exponential sofic profile, and quantum channels that need nearly linear memory

How far from finite can an amenable group be? Cornulier's sofic profile measures the least number of points on which a finite piece of a group can be modelled by permutations at accuracy 1/r. We construct a finitely presented elementary amenable group with a finite piece whose profile is exp(r^(1+o(1))): at least exp(c r/(log r)^12.75), certified by 25 explicit relations, and at most exp(C r log r). The lower bound is within a polylogarithmic factor in the exponent of the most the underlying counting criterion can give. The group places a copy of S_3 on every finite binary configuration of the three rays of Houghton's group. It embeds in Brin's higher-dimensional Thompson group 3V, which therefore has a chunk with finite, nearly exponential profile. The group defines an explicit quantum channel on C^873. A device that serves n uses of it, releasing each output before the next input arrives, needs memory about n/B when it may spend purity B, and exact devices attain this trade-off up to polylogarithmic factors. With logarithmic purity it needs memory n^(1-o(1)), and at the optimal exchange rate its memory is n^(1/2+o(1)). A local lattice process in nu dimensions whose bath starts maximally mixed needs time about delta^(-1/nu) to produce this channel to accuracy delta. A bounded time-independent Hamiltonian that is not local, coupled to a tracial bath, produces in the long run a channel with the same memory-purity trade-off. This preprint archive includes the PDF, TeX source and finite verification scripts. The scripts are regressions for the stated certificates; the general statements are proved in the manuscript. The grid variant described in a remark is only sketched.