Some Equations Over Torsion-Free Groups

Abstract

We prove that equations xm1 h1 xm2 h2 ⋯ xmk hk over torsion-free groups are solvable when k ≤ 4. In the conclusion, we assert that the Levin question is equivalent to solving certain equations with exponent sum zero over torsion-free groups.

Solving equations of length seven over torsion-free groups

“…Moreover if d(v e ) = 5 then there is a contradiction except when l(v e ) = a −1 ef −1 be. But if we assume that, subject to having a minimum number of regions, K has a maximum number of vertices of degree 2, then l(v e ) = a −1 ef −1 be does not occur since the bridge move [1] shown in Figure 3.2(v) increases the number of vertices of degree 2 (and does not change the number of regions) in K. Add c(∆) ≤ π 15 to c(∆) as shown in Figure 3 In Case 10, s 2 (t) = at 3 dt −1 et 2 gt −1 = 1 is solvable by Theorem 1 in [13].…”