Plane motion groups and virtual Poincar� duality of dimension two

Homomorphisms of nonzero degree betweenPDn-groups

“…We shall see that when n = 4 the only nontrivial edges emanating from any of the last five vertices of F 4 are (5, 4c), (5,8) and (6,7). If n > 4 then (4a), (4b), (4c), (7) and (8) are terminal vertices of F n , but there are also edges (5, 4b), (5,8), (6, 4c) and (6,7). (Also many groups in class (7) are degree 1 quotients of groups in class (5).…”

“…Let F n be the directed graph with vertices {1, 2, 3, 4a, 4b, 4c, 5, 6, 7, 8} and with an edge (i, j) if and only if i ^ j and for every group H in class (j) there is a PD n -group G in class (i) and a homomorphism of nonzero degree from G to H. Wang showed that the corresponding graph F for the fundamental groups of aspherical 3-manifolds has edges (1, n) and (2, n) for all n, (3, m) for all m > 3, (5,8) and (6,7). Moreover any homomorphism between 3-manifold groups in classes not connected by an edge in F has degree 0 [21].…”

“…The canonical epimorphism from G to G/((r, s)) = H has degree 1. Since every PD n -group of Seifert type has such a subgroup H of finite index it follows that T n has edges (3, 4b), (3, 4c), (3,5) and (3,6). When n = 3 this construction can be adapted to show that every group in class (5) or (6) is the degree 1 quotient of a group in class (3).…”

Dihedral Groups of Order 2p of Automorphisms of Compact Riemann Surfaces of Genus p – 1

“…where C is the companion matrix for Φ p (x), the cyclotomic polynomial of p-th unit roots, and U is an anti-diagonal matrix whose anti-diagonal entries are 1. Then C p = U 2 = (CU ) 2…”

“…If G is a finite group acting topologically on a surface S by orientation preserving homeomorphisms, then the positive solution of the Nielsen Realization Problem guarantees that there exists a complex analytic structure on S for which the action of G is by analytic automorphisms (see [2,4,8,10]). Thus there is no loss of generality in assuming that the action of G is complex analytic to begin with, and we will tacitly do so.…”