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The groups (2, ???? | ????, ???? | 1, ????): Finiteness and homotopy
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Abstract
We initiate the study of the groups
(
l
,
m
∣
n
,
k
∣
p
,
q
)
(l,m\mid n,k\mid p,q)
defined by the presentation
⟨
a
,
b
∣
a
l
,
b
m
,
(
a
b
)
n
,
(
a
p
b
q
)
k
⟩
\langle a,b\mid a^{l},b^{m},(ab)^{n},(a^{p}b^{q})^{k}\rangle
.
When
p
=
1
p=1
and
q
=
m
-
1
q=m-1
, we obtain the group
(
l
,
m
∣
n
,
k
)
(l,m\mid n,k)
, first systematically studied by Coxeter in 1939.
In this paper, we restrict ourselves to the case
l
=
2
l=2
and
1
n
+
1
k
≤
1
2
\frac{1}{n}+\frac{1}{k}\leq\frac{1}{2}
and give a complete determination as to which of the resulting groups are finite.
We also, under certain broadly defined conditions, calculate generating sets for the second homotopy group
π
2
(
Z
)
\pi_{2}(Z)
, where ???? is the space formed by attaching 2-cells corresponding to
(
a
b
)
n
(ab)^{n}
and
(
a
b
q
)
k
(ab^{q})^{k}
to the wedge sum of the Eilenberg–MacLane spaces ???? and ????, where
π
1
(
X
)
≅
C
2
\pi_{1}(X)\cong C_{2}
and
π
1
(
Y
)
≅
C
m
\pi_{1}(Y)\cong C_{m}
; in particular,
π
1
(
Z
)
≅
(
2
,
m
∣
n
,
k
∣
1
,
q
)
\pi_{1}(Z)\cong(2,m\mid n,k\mid 1,q)
.
Title: The groups (2, ???? | ????, ???? | 1, ????): Finiteness and homotopy
Description:
Abstract
We initiate the study of the groups
(
l
,
m
∣
n
,
k
∣
p
,
q
)
(l,m\mid n,k\mid p,q)
defined by the presentation
⟨
a
,
b
∣
a
l
,
b
m
,
(
a
b
)
n
,
(
a
p
b
q
)
k
⟩
\langle a,b\mid a^{l},b^{m},(ab)^{n},(a^{p}b^{q})^{k}\rangle
.
When
p
=
1
p=1
and
q
=
m
-
1
q=m-1
, we obtain the group
(
l
,
m
∣
n
,
k
)
(l,m\mid n,k)
, first systematically studied by Coxeter in 1939.
In this paper, we restrict ourselves to the case
l
=
2
l=2
and
1
n
+
1
k
≤
1
2
\frac{1}{n}+\frac{1}{k}\leq\frac{1}{2}
and give a complete determination as to which of the resulting groups are finite.
We also, under certain broadly defined conditions, calculate generating sets for the second homotopy group
π
2
(
Z
)
\pi_{2}(Z)
, where ???? is the space formed by attaching 2-cells corresponding to
(
a
b
)
n
(ab)^{n}
and
(
a
b
q
)
k
(ab^{q})^{k}
to the wedge sum of the Eilenberg–MacLane spaces ???? and ????, where
π
1
(
X
)
≅
C
2
\pi_{1}(X)\cong C_{2}
and
π
1
(
Y
)
≅
C
m
\pi_{1}(Y)\cong C_{m}
; in particular,
π
1
(
Z
)
≅
(
2
,
m
∣
n
,
k
∣
1
,
q
)
\pi_{1}(Z)\cong(2,m\mid n,k\mid 1,q)
.
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