### Finitely Cyclic Homogeneous Continua

Pawel Krupski, James T. Rogers
1991 Proceedings of the American Mathematical Society
A curve is finitely cyclic if and only if it is the inverse limit of graphs of genus < k , where k is some integer. In this paper it is shown that if X is a homogeneous finitely cyclic curve that is not tree-like, then X is a solenoid or X admits a decomposition into mutually homeomorphic, homogeneous, tree-like continua with quotient space a solenoid. Since the Menger curve is homogeneous, the restriction to finitely cyclic curves is essential. A continuum is a compact, connected, nonvoid
more » ... c space. A curve is a one-dimensional continuum. Each curve is an inverse limit of finite graphs. A curve is said to be arc-like if it is an inverse limit of arcs, circle-like if it is an inverse limit of circles, and tree-like if it is an inverse limit of trees. We assume that the bonding maps of an inverse sequence are surjective. A curve A is homogeneous if for each pair of points x and y of A, there exists a homeomorphism of (A, x) onto (X, y). A solenoid is an inverse limit of circles with all the bonding maps being covering maps. Under this definition, the circle is a solenoid. Each solenoid is homogeneous, since it is a topological group. A curve is called simply cyclic if it is an inverse limit of graphs each of which contains only one cycle. Hence simply cyclic curves include circle-like curves as a proper subset. The second author  has shown that if a simple cyclic, homogeneous curve is not tree-like, then either it is a solenoid or it admits a decomposition into mutually homeomorphic, tree-like, homogeneous curves with quotient space a solenoid. A curve is called finitely cyclic if it is an inverse limit of graphs of genus less than k , where k is some natural number. The purpose of this note is to prove the following theorem: Theorem. If a finitely cyclic, homogeneous curve is not tree-like, then either it is a solenoid or it admits a decomposition into mutually homeomorphic, tree-like, homogeneous curves with quotient space a solenoid.