Asphericity of a length four relative group presentation

Abstract

We consider the relative group presentation P = G, x|r where x = {x} and r = {xg 1 xg 2 xg 3 x −1 g 4 }. We show modulo a small number of exceptional cases exactly when P is aspherical. If H = g −1 1 g 2 , g −1 1 g 3 g 1 , g 4 ≤ G then the exceptional cases occur when H is isomorphic to one of C 5 , C 6 , C 8 or C 2 × C 4 . Mathematics Subject Classification: 20F05, 57M05.

On the Asphericity of a Family of Positive Relative Group Presentations

“…Let Γ (Δ,Δ) > 0 and let r be the number of corners of angle π inΔ. By Remark 2.2 (3)(i), r 1 2 n, where n = d(Δ). Set…”

“…and so c * (Δ) > 0 implies n < 12. LetΔ = (n, r) denote a region of degree n with Γ 6 = r. We need to check c * (Δ) for Δ = (n, r) = (10, 5), (10,4), (10,3), (10,2), (10,1), (8,4), (8,3), (8,2) and (8,1). The 2) only if it contains four corners with angle π (up to inversion,Δ is shown in Figure 14), and each possible l(Δ) yields a contradiction.…”

“…. x ε k g k where g i ∈ G, ε i = ±1 and g i = 1 implies ε i + ε i+1 ̸ = 0 (1 ≤ i ≤ k, subscripts mod k), then the asphericity of P has been determined (modulo some exceptional cases) when k ≤ 3 or r ∈ { xg 1 xg 2 xg 3 xg 4 , xg 1 xg 2 xg 3 x −1 g 4 , xg 1 xg 2 xg 3 xg 4 xg 5 , (xg 1 ) l 1 (xg 2 ) l 2 (xg 3 ) l 3 (l i > 1, 1 ≤ i ≤ 3)} [1][2][3] [7][8][9]. This list includes x m gx −1 h (g, h ∈ G\{1}) for m ≤ 3, and when m ≥ 4 asphericity (modulo exceptional cases) has been determined in [6].…”

“…There has been much interest in determining asphericity of P, particularly when x = {x} and r = {r} both consist of a single element. Indeed, if r = x ε1 g 1 • • • x ε k g k , where g i ∈ G, ε i = ±1 and g i = 1 implies ε i + ε i+1 = 0 (1 i k, subscripts mod k), then the asphericity of P has been determined (modulo some exceptional cases) when k 3 or r ∈ {xg 1 [1-3, 7, 8, 10]. This list includes x m gx −1 h (g, h ∈ G\{1}) for m 3, and when m 4 asphericity (modulo exceptional cases) has been determined in [6].…”

Asphericity of positive free product length 4 relative group presentations

“…Recent studies and applications of this case can be found in [7], [8] and [12]. When ε = −1 and m = 1, the asphericity of P has been determined (modulo some exceptional cases) when n = 2 in [11], when n = 3 in [1] and when n ≥ 4 in [10]. When ε = +1 and m = 1, the asphericity of P has been determined when n = 2 in [6], and (again modulo some exceptional cases)when n = 3 in [3], when n = 4 in [15] and when n ≥ 5 in [2].…”