Gaboriau's fixed price problem: do all essentially free p.m.p. actions of a countable group have the same cost?

Theorem 1.1: let $A=F(a,b_1,\dots,b_{99})$, $w=ab_1ab_2\cdots ab_{99}a$, $J=\langle b_1,\dots,b_{99},w\rangle$ and $\Gamma=A*_J(J\times\langle t\rangle)$. With $K=e^{96}96^{99}/95^{95}$, $0<\alpha<1/200$, $K\alpha^3<1/2$ and $\eta=\alpha/100$, the free part of the Bernoulli action on $[0,1]^\Gamma$ has cost at least $1+\eta$, while the skew action on $X\times\mathbb Z/M$ via $\chi(a)=1$, $\chi(b_i)=\chi(t)=0$ has cost at most $1+99/M$. Taking $M>\max\{100,99/\eta\}$ gives two essentially free p.m.p. actions of one finitely generated group with different costs. It does not address the fixed-price-one question for groups with property (T), the fixed price of any specific classical group, or the rank-gradient consequences of fixed price.

Theorem 1.1: let $A=F(a,b_1,\dots,b_{99})$, $w=ab_1ab_2\cdots ab_{99}a$, $J=\langle b_1,\dots,b_{99},w\rangle$ and $\Gamma=A*_J(J\times\langle t\rangle)$. With $K=e^{96}96^{99}/95^{95}$, $0<\alpha<1/200$, $K\alpha^3<1/2$ and $\eta=\alpha/100$, the free part of the Bernoulli action on $[0,1]^\Gamma$ has cost at least $1+\eta$, while the skew action on $X\times\mathbb Z/M$ via $\chi(a)=1$, $\chi(b_i)=\chi(t)=0$ has cost at most $1+99/M$. Taking $M>\max\{100,99/\eta\}$ gives two essentially free p.m.p. actions of one finitely generated group with different costs. It does not address the fixed-price-one question for groups with property (T), the fixed price of any specific classical group, or the rank-gradient consequences of fixed price.