Isoperimetric inequalities and the homology of groups

Algorithmically complex residually finite groups

“…For residually finite groups, the situation is different. Nilpotent groups are examples of residually finite groups with arbitrarily high polynomial Dehn function [4]. The Baumslag-Solitar groups x, y | x y = x k , k ≥ 2, are examples of residually finite (even linear) groups with exponential Dehn function.…”

“…For the Dehn function part of the theorem we use Lemma 4.20. Thus we just need to modify the Minsky machine MM 4 so that the new machine MM m , m > 4, has the same complexity of the configuration equivalence problem but time function larger than g(n 4 ).…”

“…4 where each i is a computation of MM 4 , all commands used in 1 (resp. 4 ) are new and have the same number, all commands used in 2 , 3 are old.…”

“…Consider the following new, non-deterministic Minsky machine MM 4 . Its hardware consists of the 2 glasses of MM 2 plus two more glasses, numbers 3 and 4.…”

“…Its hardware consists of the 2 glasses of MM 2 plus two more glasses, numbers 3 and 4. In every command of MM 4 we add the instruction to add a coin to glass 3 provided glass 4 is empty. Thus if a command was i; → Add(1, 2); j, we replace it by two commands i; 4 = 0 → i , i ; → Add(1, 2); j where i is a new number (not the number of any other command of the Minsky machine).…”