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Carl-Fredrik Nyberg Brodda
Carl-Fredrik Nyberg Brodda
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The answer is yes, such a group exists: this is the main result of [1]. In fact, one can take the Baumslag-Solitar group $\operatorname{BS}(p^r, -p^r) = \langle a, b \mid ba^{p^r}b^{-1} = a^{-p^r} \rangle$ where $p$ is an odd prime and $r \geq 1$. These are residually nilpotent, but are not residually a finite $q$-group for any prime $q$.

In particular, the smallest example of this form is $\operatorname{BS}(3, -3) = \langle a, b \mid ba^3b^{-1}a^3 = 1 \rangle$.

(Note that OP's question was posed by McCarron [2] in 1996).

${}$

[1] Moldavanskii, D. I., Residual nilpotence of groups with one defining relation, +; translation from Mat. Zametki 107, No. 5, 752-759 (2020). ZBL07215590.

[2] McCarron, James, Residually nilpotent one-relator groups with non-trivial centre, Proc. Amer. Math. Soc. 124, No. 1 (1996).

The answer is yes, such a group exists: this is the main result of [1]. In fact, one can take the Baumslag-Solitar group $\operatorname{BS}(p^r, -p^r) = \langle a, b \mid ba^{p^r}b^{-1} = a^{-p^r} \rangle$ where $p$ is an odd prime and $r \geq 1$. These are residually nilpotent, but are not residually a finite $q$-group for any prime $q$.

In particular, the smallest example of this form is $\operatorname{BS}(3, -3) = \langle a, b \mid ba^3b^{-1}a^3 = 1 \rangle$.

(Note that OP's question was posed by McCarron [2] in 1996).

${}$

[1] Moldavanskii, D. I., Residual nilpotence of groups with one defining relation, +; translation from Mat. Zametki 107, No. 5, 752-759 (2020). ZBL07215590.

[2] McCarron, James, Residually nilpotent one-relator groups with non-trivial centre, Proc. Amer. Math. Soc. 124, No. 1 (1996).

The answer is yes, such a group exists: this is the main result of [1]. In fact, one can take the Baumslag-Solitar group $\operatorname{BS}(p^r, -p^r) = \langle a, b \mid ba^{p^r}b^{-1} = a^{-p^r} \rangle$ where $p$ is an odd prime and $r \geq 1$. These are residually nilpotent, but are not residually a finite $q$-group for any prime $q$.

In particular, the smallest example of this form is $\operatorname{BS}(3, -3) = \langle a, b \mid ba^3b^{-1}a^3 = 1 \rangle$.

(Note that OP's question was posed by McCarron [2] in 1996).

${}$

[1] Moldavanskii, D. I., Residual nilpotence of groups with one defining relation, translation from Mat. Zametki 107, No. 5, 752-759 (2020). ZBL07215590.

[2] McCarron, James, Residually nilpotent one-relator groups with non-trivial centre, Proc. Amer. Math. Soc. 124, No. 1 (1996).

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Carl-Fredrik Nyberg Brodda
Carl-Fredrik Nyberg Brodda
  • 6.5k
  • 4
  • 27
  • 62

The answer is yes, such a group exists: this is the main result of [1]. In fact, one can take the Baumslag-Solitar group $\operatorname{BS}(p^r, -p^r) = \langle a, b \mid ba^{p^r}b^{-1} = a^{-p^r} \rangle$ where $p$ is an odd prime and $r \geq 1$. These are residually nilpotent, but are not residually a finite $q$-group for any prime $q$.

In particular, the smallest example of this form is $\operatorname{BS}(3, -3) = \langle a, b \mid ba^3b^{-1}a^3 = 1 \rangle$.

(Note that OP's question was posed by McCarron [2] in 1996).

${}$

[1] Moldavanskii, D. I., Residual nilpotence of groups with one defining relation, +; translation from Mat. Zametki 107, No. 5, 752-759 (2020). ZBL07215590.

[2] McCarron, James, Residually nilpotent one-relator groups with non-trivial centre, Proc. Amer. Math. Soc. 124, No. 1 (1996). 

The answer is yes, such a group exists: this is the main result of [1]. In fact, one can take the Baumslag-Solitar group $\operatorname{BS}(p^r, -p^r) = \langle a, b \mid ba^{p^r}b^{-1} = a^{-p^r} \rangle$ where $p$ is an odd prime and $r \geq 1$. These are residually nilpotent, but are not residually a finite $q$-group for any prime $q$.

In particular, the smallest example of this form is $\operatorname{BS}(3, -3) = \langle a, b \mid ba^3b^{-1}a^3 = 1 \rangle$.

(Note that OP's question was posed by McCarron in 1996).

${}$

[1] Moldavanskii, D. I., Residual nilpotence of groups with one defining relation, +; translation from Mat. Zametki 107, No. 5, 752-759 (2020). ZBL07215590.

The answer is yes, such a group exists: this is the main result of [1]. In fact, one can take the Baumslag-Solitar group $\operatorname{BS}(p^r, -p^r) = \langle a, b \mid ba^{p^r}b^{-1} = a^{-p^r} \rangle$ where $p$ is an odd prime and $r \geq 1$. These are residually nilpotent, but are not residually a finite $q$-group for any prime $q$.

In particular, the smallest example of this form is $\operatorname{BS}(3, -3) = \langle a, b \mid ba^3b^{-1}a^3 = 1 \rangle$.

(Note that OP's question was posed by McCarron [2] in 1996).

${}$

[1] Moldavanskii, D. I., Residual nilpotence of groups with one defining relation, +; translation from Mat. Zametki 107, No. 5, 752-759 (2020). ZBL07215590.

[2] McCarron, James, Residually nilpotent one-relator groups with non-trivial centre, Proc. Amer. Math. Soc. 124, No. 1 (1996). 

Source Link
Carl-Fredrik Nyberg Brodda
Carl-Fredrik Nyberg Brodda
  • 6.5k
  • 4
  • 27
  • 62

The answer is yes, such a group exists: this is the main result of [1]. In fact, one can take the Baumslag-Solitar group $\operatorname{BS}(p^r, -p^r) = \langle a, b \mid ba^{p^r}b^{-1} = a^{-p^r} \rangle$ where $p$ is an odd prime and $r \geq 1$. These are residually nilpotent, but are not residually a finite $q$-group for any prime $q$.

In particular, the smallest example of this form is $\operatorname{BS}(3, -3) = \langle a, b \mid ba^3b^{-1}a^3 = 1 \rangle$.

(Note that OP's question was posed by McCarron in 1996).

${}$

[1] Moldavanskii, D. I., Residual nilpotence of groups with one defining relation, +; translation from Mat. Zametki 107, No. 5, 752-759 (2020). ZBL07215590.