# Are these two kernels isomorphic groups?

MathOverflow Asked by Francesco Polizzi on January 12, 2021

We have a finitely presented, infinite group $$mathsf{B}$$,
coming from a geometric topology problem (it is the quotient of a braid group for a genus 2 surface).

It is generated by elements
$$begin{equation} {r_{11}, , t_{11}, , r_{12}, , t_{12}, , r_{21}, , t_{21}, , r_{22}, , t_{22}, , z }, end{equation}$$
subject to the following relations (here $$[a, , b]=aba^{-1}b^{-1}$$):
begin{equation*} begin{aligned} & z^2=1 \ & [r_{12}^{-1}, , t_{12}^{-1}] , t_{12}^{-1} , [r_{11}^{-1}, , t_{11}^{-1}] , t_{11}^{-1}, (t_{11} , t_{12}) = z & \ & [r_{21}^{-1}, , t_{21}] ; t_{21} ; [r_{22}^{-1}, , t_{22}] , t_{22}, (t_{22}^{-1} , t_{21}^{-1})= z^{-1} \ & \ & [r_{11}, , r_{22}]=1 & & [r_{12}, , r_{22}]=1 \ & [r_{11}, , r_{21}]=1 & & [r_{12}, , r_{21}]= z^{-1},r_{21},r_{22}^{-1},z,r_{22},r_{21}^{-1} \ & [r_{11}, , t_{22}]=1 & & [r_{12}, , t_{22}]=z^{-1} \ & [r_{11}, , t_{21}]=z^{-1} & & [r_{12}, , t_{21}]=[z^{-1}, , t_{21}] \ & [r_{11}, , z]=[r_{21}^{-1}, , z] & & [r_{12}, , z]=[r_{22}^{-1}, , z] \ & \ & [t_{11}, , r_{22}]=1 & & [t_{12}, , r_{22}]= t_{22}^{-1}, z , t_{22} \ & [t_{11}, , r_{21}]=t_{21}^{-1}, z , t_{21} & & [t_{12}, , r_{21}]= [t_{22}^{-1}, , z] \ & [t_{11}, , t_{22}]=1 & & [t_{12}, , t_{22}]=[t_{22}^{-1}, , z] \ & [t_{11}, , t_{21}]=[t_{21}^{-1}, , z] & & [t_{12}, , t_{21}]= t_{22}^{-1}, z , t_{22} , z^{-1} , t_{21} , z , t_{22}^{-1},z^{-1} ,t_{22},t_{21}^{-1} \ & [t_{11}, , z]=[t_{21}^{-1}, , z] & & [t_{12}, , z]=[t_{22}^{-1}, , z] \ end{aligned} end{equation*}
Next, we consider the two extra-special groups of order $$32$$, whose labels are $$[32, , 49]$$ and $$[32, , 50]$$ in GAP4 notation, and whose presentations are as follows:
$$begin{equation} begin{split} mathsf{G}(32, , 49) = langle , & mathsf{r}_1, , mathsf{t}_1, , mathsf{r}_2,, mathsf{t}_2, , mathsf{z} ; | ; mathsf{r}_{j}^2 = mathsf{t}_{j}^2=mathsf{z}^2=1, \ & [mathsf{r}_{j}, , mathsf{z}] = [mathsf{t}_{j}, , mathsf{z}]= 1, \ & [mathsf{r}_j, mathsf{r}_k]= [mathsf{t}_j, mathsf{t}_k] = 1, \ & [mathsf{r}_{j}, ,mathsf{t}_{k}] =mathsf{z}^{- delta_{jk}} , rangle end{split} end{equation}$$
$$begin{equation} begin{split} mathsf{G}(32, , 50) = langle , & mathsf{r}_1, , mathsf{t}_1, , mathsf{r}_2,, mathsf{t}_2, , mathsf{z} ; | ; mathsf{r}_{1}^2 = mathsf{t}_{1}^2=mathsf{z}^2=1, ; mathsf{r}_{2}^2 = mathsf{t}_{2}^2=mathsf{z}\ & [mathsf{r}_{j}, , mathsf{z}] = [mathsf{t}_{j}, , mathsf{z}]= 1, \ & [mathsf{r}_j, mathsf{r}_k]= [mathsf{t}_j, mathsf{t}_k] = 1, \ & [mathsf{r}_{j}, ,mathsf{t}_{k}] =mathsf{z}^{- delta_{jk}} , rangle end{split} end{equation}$$
There are two surjective group homomorphisms $$varphi_{49} colon mathsf{B} to mathsf{G}(32, , 49), quad varphi_{50} colon mathsf{B} to mathsf{G}(32, , 50),$$ expressed in both cases by
$$[r_{11}, ; t_{11}, ; r_{12}, ; t_{12}, ; r_{21}, ; t_{21}, ; r_{22}, ; t_{22}, ; z] mapsto [mathsf{t_2}, ; mathsf{t}_1, ; mathsf{t}_1 mathsf{r}_2, ; mathsf{r}_1, ; mathsf{t}_2 mathsf{r}_1, ; mathsf{r}_2, ; mathsf{t}_1, ; mathsf{t}_2, ; mathsf{z}].$$
Let us denote their kernels by $$mathsf{K}_{49}$$ and $$mathsf{K}_{50}$$, respectively.

Question. Are $$mathsf{K}_{49}$$ and $$mathsf{K}_{50}$$ isomorphic groups?

I have constructed $$mathsf{K}_{49}$$ and $$mathsf{K}_{50}$$ by using GAP4, and I tried to make some calculations on them. Unfortunately, GAP4 does not work with infinite groups so well as it works with finite groups, so the computational approach seems to be difficult (at least for me). The usual invariants that I was able to compute ($$mathtt{AbelianInvariants}$$, $$mathtt{MaximalAbelianQuotient})$$ were not useful in order to distinguish them.

Edit: Here is the GAP4 code I used in order to construct the group $$mathsf{B}$$ and the two kernels.

ColorPrompt(true);

#redefine commutators
comm:=function(x, y) return x*y*x^-1*y^-1; end;

#Construction of the Group B
F:=FreeGroup("r11", "t11", "r12", "t12", "r21", "t21", "r22", "t22", "z");;
r11:=F.1;; t11:=F.2;; r12:=F.3;; t12:=F.4;;
r21:=F.5;; t21:=F.6;; r22:=F.7;; t22:=F.8;; z:=F.9;;

U:=z^2;;
S1:=comm(r12^(-1), t12^(-1))*t12^(-1)*comm(r11^(-1), t11^(-1))*t12*z^(-1);;
S2:=comm(r21^(-1), t21)*t21*comm(r22^(-1), t22)*t21^(-1)*z;;
R1:=comm(r11, r22);;
R2:=comm(r11, r21);;
R3:=comm(r11, t22);;
R4:=comm(r11, t21)*z;;
R5:=comm(r11, z)*comm(z, r21^(-1));;
R6:=comm(r12, r22);;
R7:=z^(-1)*r21*r22^(-1)*z*r22*r21^(-1)*comm(r21, r12);;
R8:=comm(r12, t22)*z;;
R9:=comm(r12, t21)*comm(t21, z^(-1));;
R10:=comm(r12, z)*comm(z, r22^(-1));;
T1:=comm(t11, r22);;
T2:=comm(t11, r21)*t21^(-1)*z^(-1)*t21;;
T3:=comm(t22, t11);;
T4:=comm(t11, t21)*comm(z, t21^(-1));;
T5:=comm(t11, z)*comm(z, t21^(-1));;
T6:=comm(t12, r22)*t22^(-1)*z^(-1)*t22;;
T7:=comm(t12, r21)*comm(z, t22^(-1));;
T8:=comm(t12, t22)*comm(z, t22^(-1));;
T9:=t22^(-1)*z*t22*z^(-1)*t21*z*t22^(-1)*z^(-1)*t22*t21^(-1)*comm(t21, t12);;
T10:=comm(t12, z)*comm(z, t22^(-1));;

B:=F/[U, S1, S2, R1, R2, R3, R4, R5, R6, R7, R8, R9, R10, T1, T2, T3, T4, T5, T6, T7, T8, T9, T10];;
r11:=B.1;; t11:=B.2;; r12:=B.3;; t12:=B.4;;
r21:=B.5;; t21:=B.6;; r22:=B.7;; t22:=B.8;; z:=B.9;;

#construction of G(32, 49)
f:=FreeGroup("r1", "t1", "r2", "t2", "s");
r1:=f.1;; t1:=f.2;; r2:=f.3;; t2:=f.4;; s:=f.5;;
g:=f/[r1^2, t1^2, r2^2, t2^2, s^2, comm(r1, s), comm(t1, s), comm(r2, s), comm(t2,s),
comm(r1, r2), comm(t1, t2), comm(r1, t1)*s, comm(r1, t2), comm(r2, t1), comm(r2, t2)*s];;
r1:=g.1;; t1:=g.2;; r2:=g.3;; t2:=g.4;; s:=g.5;;
IdSmallGroup(g); StructureDescription(g);

#construction of the group homomorphism B->G(32, 49)
hom49:=GroupHomomorphismByImages(B,g,[r11, t11, r12, t12, r21, t21, r22, t22, z],
[t2, t1, r2*t1, r1, r1*t2, r2, t1, t2, s]);
K49:=Kernel(hom49);

#construction of G(32, 50);
o:=FreeGroup("u1", "v1", "u2", "v2", "y");
u1:=o.1;; v1:=o.2;; u2:=o.3;; v2:=o.4;; y:=o.5;;
h:=o/[u1^2, v1^2, u2^2*(y^-1), v2^2*(y^-1), y^2, comm(u1, y), comm(v1, y), comm(u2, y), comm(v2,y), comm(u1, u2), comm(v1, v2), comm(u1, v1)*y, comm(u1, v2), comm(u2, v1), comm(u2, v2)*y];;
u1:=h.1;; v1:=h.2;; u2:=h.3;; v2:=h.4;; y:=h.5;;
IdSmallGroup(h); StructureDescription(h);

#construction of the group epimomorphism B->G(32, 50)
hom50:=GroupHomomorphismByImages(B,h,[r11, t11, r12, t12, r21, t21, r22, t22, z],
[v2, v1, u2*v1, u1, u1*v2, u2, v1, v2, y]);
K50:=Kernel(hom50);

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