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∣al,bm,(a⁢b)n,(ap⁢bq)k⟩\langle a,b\mid a^{l},b^{m},(ab)^{n},(a^{p}b^{q})^{k}\rangle. When p=1p=1 and q=m-1q=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=2l=2 and 1n+1k≤12\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 π2⁢(Z)\pi_{2}(Z), where 𝑍 is the space formed by attaching 2-cells corresponding to (a⁢b)n(ab)^{n} and (a⁢bq)k(ab^{q})^{k} to the wedge sum of the Eilenberg–MacLane spaces 𝑋 and 𝑌, where π1⁢(X)≅C2\pi_{1}(X)\cong C_{2} and π1⁢(Y)≅Cm\pi_{1}(Y)\cong C_{m}; in particular, π1(Z)≅(2,m∣n,k∣1,q)\pi_{1}(Z)\cong(2,m\mid n,k\mid 1,q).

中文翻译：

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组（2，𝑚|𝑛，𝑘| 1，𝑞）：有限性和同伦性

我们开始研究由表示⟨a，b∣al，bm，（a）定义的组（l，m∣n，k∣p，q）（l，m \ mid n，k \ mid p，q） ⁢b）n，（ap⁢bq）k⟩\ langle a，b \ mid a ^ {l}，b ^ {m}，（ab）^ {n}，（a ^ {p} b ^ {q} ）^ {k} \ rangle。当p = 1p = 1且q = m-1q = m-1时，我们得到（l，m∣n，k）（l，m \ mid n，k），这是1939年由Coxeter首次系统地研究的。在本文中，我们将自己限制为l = 2l = 2且1n +1k≤12\ frac {1} {n} + \ frac {1} {k} \ leq \ frac {1} {2}完全确定哪个结果组是有限的。我们还可以在某些广义定义的条件下，计算第二同伦群π2⁢（Z）\ pi_ {2}（Z）的生成集，其中𝑍是通过附加与（a⁢b）n对应的2个单元格形成的空间（ab）^ {n}和（a⁢bq）k（ab ^ {q}）^ {k}到Eilenberg–MacLane空间𝑋和the的楔形和，其中π1⁢（X）≅C2\ pi_ {1}（X）\ cong C_ {2}和π1⁢（Y）≅Cm\ pi_ {1}（Y）\ cong C_ {m}; 尤其是π1（Z）≅（2，m∣n，k∣1，q）\ pi_ {1}（Z）\ cong（2，m \ mid n，k \ mid 1，q）。