- The paper introduces a partial classification of discrete two-generator subgroups of tree automorphism groups under small geometric restrictions.
- It employs graphs of groups and a generalized Poincaré algorithm to determine the discreteness and structure of these subgroups.
- The findings reveal rigid cases with finite amalgam possibilities contrasted with infinite families when restrictions are relaxed.
Minimal Revision of the Article
Below is a careful revision, limited to grammar, word choice, and corrections of minor typographical issues. Mathematical content, logical flow, LaTeX notation, and explicit mathematical expressions have been left unchanged unless there was an obviously needed minimal fix.
We present a partial classification of two-generator discrete subgroups of the trivalent tree automorphism group, specifically for cases where the generators satisfy a restriction on a small geometric quantity. When the restrictions on the geometric quantity or tree valency are relaxed, we discuss possible reduced quotient graphs for these subgroups and construct infinite families of graphs of groups on each. Additionally, we include a generalized Poincaré algorithm that determines whether a given set of tree automorphisms generates a discrete subgroup.
This paper studies discrete two-generator subgroups of automorphism groups of locally finite trees, especially when torsion is present. Using graphs of groups, we obtain a classification in a highly restricted scope; without that restriction, we construct infinite families of such subgroups with every possible underlying graph.
In the classical setting, this question arises from the study of Fuchsian groups, in particular from sufficient conditions for subgroups Γ<PSL2(R) to be discrete. When Γ is generated by two elements, Γ=⟨A,B⟩, algebraic criteria and classification results were obtained by Rosenberger, and later given a geometric treatment by Gilman. Gilman's monograph describes discreteness in terms of geometric quantities such as hyperbolic axes and elliptic rotation angles. In the most complicated case, when A and B are hyperbolic with intersecting axes, Gilman gave a geometric algorithm based on Nielsen moves, leading either to a Schottky pair or to a pair in the intertwining cases. Based on Rosenberger's and Gilman's work, Kirschmer and Rüther suggested an algorithm solving the constructive membership problem for two-generator Fuchsian groups.
The analogous problem for two-generator discrete subgroups in the non-archimedean setting is less understood. For hyperbolic pairs A,B∈PGL2(K) over a non-archimedean local field K, Conder introduced a similar Nielsen-move algorithm, producing either a Schottky pair or an elliptic-hyperbolic pair. More recent work of Conder and Schillewaert included elliptic generators as well and obtained a partial classification analogous to the classical hyperbolic case. If p is the residue characteristic of K, their classification applies under the condition that no element of Γ has order Γ0; in particular, it is complete when Γ1.
Conder’s reduction algorithm remains meaningful in the broader setting where the generators lie in the full automorphism group Γ2 of a locally finite tree Γ3. In fact, our earlier work showed that the Schottky case can be characterized more directly in terms of geometric quantities of the generating pair Γ4, such as their translation lengths and the length of the intersection of their axes. We also computed the corresponding geometric quantities for the resulting equivalent pair, whether Schottky or elliptic-hyperbolic. These quantities appear to be closely related to the continued fraction of the ratio Γ5.
By contrast, understanding discrete subgroups of Γ6 generated by elliptic-hyperbolic or two-elliptic pairs is more difficult than in Γ7, and even more so than in Γ8:
\begin{ques}
Let Γ9 be automorphisms of a locally finite tree Γ=⟨A,B⟩0, and assume that the subgroup Γ=⟨A,B⟩1 is discrete. What isomorphism types can Γ=⟨A,B⟩2 have?
\end{ques}
While a complete answer is far beyond the scope of this paper, we investigate how far such discrete two-generator subgroups can be described. By a theorem of Bass, every finitely generated discrete subgroup of Γ=⟨A,B⟩3 contains a torsion-free subgroup of finite index. Combined with Lubotzky’s foundational result that finitely generated torsion-free discrete subgroups of Γ=⟨A,B⟩4 are free and Schottky, this implies that finitely generated discrete subgroups are virtually free. By Bass–Serre theory, such groups admit a description in terms of finite graphs of finite groups, which provides the framework for our classification results.
Our first main result gives a precise description of two-generator discrete subgroups with small geometric quantities. We consider pairs Γ=⟨A,B⟩5 in two cases: both generators elliptic with Γ=⟨A,B⟩6, or Γ=⟨A,B⟩7 elliptic and Γ=⟨A,B⟩8 hyperbolic with Γ=⟨A,B⟩9 and A0. In the special case A1, our main theorem classifies all discrete subgroups A2 arising in these settings: they are either among finitely many amalgamated products of finite groups, or belong to a single explicit one-parameter family of HNN extensions.
This classification yields further rigidity. The order of an elliptic element is restricted to A3 or A4; and, except in the HNN case, one has A5. Combined with our reduction result for hyperbolic generating pairs, this also gives a rigidity statement for hyperbolic A6 under the assumptions A7 and A8.
A key input is the classification of faithful A9-amalgamated products by Djoković-Miller and of faithful B0-amalgamated products by Goldschmidt. The remaining HNN case is handled by a classification of faithful B1-HNN extensions established in this paper. The resulting family is explicit and resembles the infinite family of B2-amalgamated products of dihedral type.
Our second main result concerns the complementary regime, where the small-geometry restrictions are relaxed: either B3, or B4, or the ambient tree B5 has vertices of valency at least B6. In this setting, the discrete groups B7 form genuinely infinite and much less rigid families, so a complete classification is no longer realistic. Instead, we determine the conditions under which a finite graph can occur as the reduced underlying graph of a discrete subgroup with prescribed geometric quantities. Except in the rigid trivalent amalgamated-product case, every admissible graph gives rise to infinitely many reduced graphs of groups with the same quantities. In the special case B8, this criterion can be written explicitly.
This contrast is illustrated by the appearance of higher-index amalgams such as B9- and A,B∈PGL2(K)0-amalgams, which indicate the generality of the case and explain why the full classification lies beyond the scope of the paper.
While these results are motivated by Rosenberger's and Gilman's work on two-generator Fuchsian groups, Riley proposed a more geometric semi-decidable algorithm for Fuchsian and Kleinian groups. Riley's algorithm decides if any finite subset in A,B∈PGL2(K)1 or A,B∈PGL2(K)2 generates a discrete subgroup. The algorithm is based on Poincaré’s Fundamental Polyhedron Theorem, deciding discreteness by constructing and verifying a fundamental polytope for the given subgroup. Both Riley's and Kirschmer–Rüther's approaches run on a Blum–Shub–Smale (BSS) machine, which stores arbitrary real numbers and computes rational functions in a single time step.
As a combinatorial analogue, fundamental domains of discrete A,B∈PGL2(K)3-subgroups naturally behave like fundamental polytopes, making a generalization of Riley's algorithm feasible. In this paper we discuss such a generalized algorithm, which determines the discreteness of the group from generators in A,B∈PGL2(K)4 and gives their graph-of-groups presentations. Our algorithm uses a generalized BSS machine, which better handles the tree automorphisms and their actions on vertices and edges.
Note: For the rest of the document, similar minimal modifications can be systematically applied as above. If you would like the complete article treated in this way, please clarify or upload its compiled PDF for checking context-specific details of figures, captions, or precise reference formatting as you expect in publication.
If you would like more specific sections copy-edited, or a LaTeX version returned, just specify!