Return to Answer

http -> https

Source Link

edited Dec 4, 2022 at 1:22

4.7k
4
35
40

The groups have trivial center, as pointed out by Sam Nead. Another, more combinatorial, way to show this is to apply the algorithm from [1], which decides whether any given one-relator group has a non-trivial center or not. If one wishes to be very explicit, one can even compute an explicit generating set for a finite index surface group inside the groups, see [2], particularly §5, where the Reidemeister-Schreier method is used to find a one-relator presentation on $4q-2$ generators for such a finite index subgroup.

${}$

${}$

[1] Baumslag, G.; Taylor, Tekla, The centre of groups with one defining relatorThe centre of groups with one defining relator, Math. Ann. 175, 315-319 (1968). ZBL0157.34901.

[2] Baumslag, Gilbert; Troeger, Douglas, Virtually free-by-cyclic one-relator groups. I., Fine, Benjamin (ed.) et al., Aspects of infinite groups. A Festschrift in honor of Anthony Gaglione. Papers of the conference, Fairfield, USA, March 2007 in honour of Anthony Gaglione’s 60th birthday. Hackensack, NJ: World Scientific (ISBN 978-981-279-340-9/hbk). Algebra and Discrete Mathematics (Hackensack) 1, 9-25 (2008). ZBL1188.20023.

The groups have trivial center, as pointed out by Sam Nead. Another, more combinatorial, way to show this is to apply the algorithm from [1], which decides whether any given one-relator group has a non-trivial center or not. If one wishes to be very explicit, one can even compute an explicit generating set for a finite index surface group inside the groups, see [2], particularly §5, where the Reidemeister-Schreier method is used to find a one-relator presentation on $4q-2$ generators for such a finite index subgroup.

${}$

${}$

[1] Baumslag, G.; Taylor, Tekla, The centre of groups with one defining relator, Math. Ann. 175, 315-319 (1968). ZBL0157.34901.

[2] Baumslag, Gilbert; Troeger, Douglas, Virtually free-by-cyclic one-relator groups. I., Fine, Benjamin (ed.) et al., Aspects of infinite groups. A Festschrift in honor of Anthony Gaglione. Papers of the conference, Fairfield, USA, March 2007 in honour of Anthony Gaglione’s 60th birthday. Hackensack, NJ: World Scientific (ISBN 978-981-279-340-9/hbk). Algebra and Discrete Mathematics (Hackensack) 1, 9-25 (2008). ZBL1188.20023.

The groups have trivial center, as pointed out by Sam Nead. Another, more combinatorial, way to show this is to apply the algorithm from [1], which decides whether any given one-relator group has a non-trivial center or not. If one wishes to be very explicit, one can even compute an explicit generating set for a finite index surface group inside the groups, see [2], particularly §5, where the Reidemeister-Schreier method is used to find a one-relator presentation on $4q-2$ generators for such a finite index subgroup.

${}$

${}$

[1] Baumslag, G.; Taylor, Tekla, The centre of groups with one defining relator, Math. Ann. 175, 315-319 (1968). ZBL0157.34901.

[2] Baumslag, Gilbert; Troeger, Douglas, Virtually free-by-cyclic one-relator groups. I., Fine, Benjamin (ed.) et al., Aspects of infinite groups. A Festschrift in honor of Anthony Gaglione. Papers of the conference, Fairfield, USA, March 2007 in honour of Anthony Gaglione’s 60th birthday. Hackensack, NJ: World Scientific (ISBN 978-981-279-340-9/hbk). Algebra and Discrete Mathematics (Hackensack) 1, 9-25 (2008). ZBL1188.20023.

Source Link

created Nov 21, 2022 at 17:29

Carl-Fredrik Nyberg Brodda

6.5k
4
27
62

The groups have trivial center, as pointed out by Sam Nead. Another, more combinatorial, way to show this is to apply the algorithm from [1], which decides whether any given one-relator group has a non-trivial center or not. If one wishes to be very explicit, one can even compute an explicit generating set for a finite index surface group inside the groups, see [2], particularly §5, where the Reidemeister-Schreier method is used to find a one-relator presentation on $4q-2$ generators for such a finite index subgroup.

${}$

${}$

[1] Baumslag, G.; Taylor, Tekla, The centre of groups with one defining relator, Math. Ann. 175, 315-319 (1968). ZBL0157.34901.

[2] Baumslag, Gilbert; Troeger, Douglas, Virtually free-by-cyclic one-relator groups. I., Fine, Benjamin (ed.) et al., Aspects of infinite groups. A Festschrift in honor of Anthony Gaglione. Papers of the conference, Fairfield, USA, March 2007 in honour of Anthony Gaglione’s 60th birthday. Hackensack, NJ: World Scientific (ISBN 978-981-279-340-9/hbk). Algebra and Discrete Mathematics (Hackensack) 1, 9-25 (2008). ZBL1188.20023.