All generating pairs of all two-generator Fuchsian groups

Primitive, Almost Primitive, Test, and Δ-Primitive Elements of Free Algebras with the Nielsen–Schreier Property

A test element in a group G is an element g with the property that if f(g) = g for an endomorphism f of G to G then f must be an automorphism. A test element in a free group is called a test word. Nielsen gave the first example of a test word by showing that in the free group on x, y the commutator [x, y] satisfies this property. T. Turner recently characterized test words as those elements of a free group contained in no proper retract. Since free factors are retracts, test words are therefore very strong forms of non-primitive elements. In this paper we give some new examples of test words and examine the relationship between test elements and several other concepts, in particular generic elements and almost-primitive elements (APE’s). In particular we show that an almost primitive element which lies in a certain type of verbal subgroup must be a test word. Further using a theorem of Rosenberger on equations in free products we prove a result on APE’s, generic elements and test words in certain free products of free groups. Finally we examine test elements in non-free groups and introduce the concept of the test rank of a group.

https://msp.org/pjm/1999/190-2/pjm-v190-n2-p06-s.pdf

Test words, generic elements and almost primitivity

Commutators, generators and conjugacy equations in groups

In (12), Takeuchi characterized those Fuchsian groups which are arithmetic in terms of the traces of the matrices involved in the group. Further, he used these criteria (13) to show that there are only finitely many arithmetic Fuchsian triangle groups and indeed to determine the signatures of all such groups. We note that, if two Fuchsian triangle groups have the same signature, then they are conjugate in PGL(2, ) and hence either all Fuchsian triangle groups of a given signature are arithmetic or none of them are. This will not, in general, be the case for non-triangle Fuchsian groups and in this paper, we examine Fuchsian groups with signature of one of the forms (1; n; 0), n ≽ 2 or (0; 2,2, 2, n; 0), n ≽ 3 and odd in which case the space of conjugacy classes has dimension two. Our principal results state that for each n, there are only finitely many conjugacy classes of arithmetic Fuchsian groups of given signature (1; n; 0) or (0; 2, 2, 2, n; 0) and for large enough n there are no arithmetic Fuchsian groups of that given signature (Theorems 3 and 4). Together with the results of (13) and known results for non-cocompact Fuchsian groups, these results show that there are only finitely many conjugacy classes of two-generator arithmetic Fuchsian groups (see §7).

Two-generator arithmetic Fuchsian groups

(1978). Automorphisms of the fuchsian groups op type (0;2,2,2,q;0) Communications in Algebra: Vol. 6, No. 11, pp. 1115-1129.

Gerhard Rosenberge

Papers

Automorphisms of the fuchsian groups op type (0;2,2,2,q;0)