By Gilman R.H., Hermiller S., Holt D.F.
We end up finitely generated team G is almost unfastened if and provided that there exists a producing set for G and к > zero such that every one k-locally geodesic phrases with admire to that producing set are geodesic.
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Extra info for A characterisation of virtually free groups
B) Their solutions represent the ∞2 above chosen integral manifolds z1 = f, z2 = ϕ of the involutory system F1 = 0, F2 = 0, F3 = 0. 27 General theory of partial differential equations The form of the equations (B) depends more or less upon choice of integral manifolds of the involutory system. However, it is possible to find a first-order partial differential equation of an absolutely definite form, satisfied by all integral manifolds of the involutory system. Indeed, there exist three equations Fk (x, y, f, ϕ, fx , ϕx , fy , ϕy ) = 0, satisfied identically in x, y, a, and b, for which we have already selected ∞2 integral manifolds of the involutory system.
Xn . Let us denote ∂zi (i) · pk = ∂xk and write the system in the form (1) Fν (x1 , . . , xn , z1 , . . , zm , p1 , . . , p(m) n ) = 0 (ν = 1, 2, . ). (l) (m) Every system of quantities x1 , . . , xn , z1 , . . , zm , p1 , . . , pn is termed an element. Two neighboring elements(26) are said to be united if they satisfy the system of equations (i) dzi − p1 dx1 − . . − p(i) n dxn = 0 (i = 1, . . , m). A set of united elements is termed a union of elements(27) . A union of elements contains maximum ∞n and minimum ∞1 elements.
Moreover, the three values x, y, z can also have constant values not for every characteristic strip. This is provided by the fact that the number of existing two-dimensional integral manifolds is at least ∞4 and that the set of all two-dimensional manifolds does not satisfy any first-order partial differential equation other than Ω = 0 so that every characteristic strip belongs at least to one two-dimensional manifold z = f, p = fx , q = fy . It follows that if x, y, z or only x, y have constant values for a characteristic strip, then z, p, q are also constant for the same strip.
A characterisation of virtually free groups by Gilman R.H., Hermiller S., Holt D.F.