I apologize if this is too obvious, but I figure it must have a quick answer.
Are there open subgroups $\Gamma\le SL_2(\widehat{\mathbb{Z}})$ which are conjugate in $GL_2(\widehat{\mathbb{Z}})$, but not conjugate in $SL_2(\widehat{\mathbb{Z}})$?
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Sign up to join this communityI apologize if this is too obvious, but I figure it must have a quick answer.
Are there open subgroups $\Gamma\le SL_2(\widehat{\mathbb{Z}})$ which are conjugate in $GL_2(\widehat{\mathbb{Z}})$, but not conjugate in $SL_2(\widehat{\mathbb{Z}})$?
Yes, there are indeed such subgroups. Since the open subgroups are exactly the congruence subgroups, it suffices to find two subgroups of $\mathrm{SL}_2(\mathbf{Z}/N\mathbf{Z})$ which are conjugate in $\mathrm{GL}_2(\mathbf{Z}/N\mathbf{Z})$ but not in $\mathrm{SL}_2(\mathbf{Z}/N\mathbf{Z})$. The following Magma code shows that the first example arises for $N=7$:
for N:=2 to 10 do
G:=GL(2,IntegerRing(N));
H:=SL(2,IntegerRing(N));
Sub:=Subgroups(H);
for R1,R2 in Sub do
H1:=R1`subgroup;
H2:=R2`subgroup;
if H1 ne H2 then
bool,g:=IsConjugate(G,H1,H2);
if bool then print H1,H2,g; end if;
end if;
end for;
end for;
whose first output is
MatrixGroup(2, IntegerRing(7)) of order 2^3
Generators:
[2 1]
[2 5]
[6 4]
[3 1]
[6 0]
[0 6]
MatrixGroup(2, IntegerRing(7)) of order 2^3
Generators:
[1 1]
[5 6]
[6 4]
[3 1]
[6 0]
[0 6]
Conjugate by
[3 0]
[0 1]