REVIEW 2 major objections 4 minor 20 references
Discrete and free two-generated subgroups of ${\rm SL_2}$ over non-archimedean local fields
T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read A finite algorithm decides whether any two matrices over a non-archimedean local field generate a discrete free subgroup of rank two.
desk verdict A genuinely useful algorithm for detecting discrete free rank-two subgroups over non-archimedean local fields, with a solid erratum to Paulin's 1989 formula; the only soft spot is a terse midpoint check that is easily patched. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing mechanism is translation-length-minimizing Nielsen transformations. A Nielsen transformation replaces the two generators by another pair generating the same subgroup, such as swapping them or replacing one by $X^{-1}Y$; here the transformations are chosen to decrease the ordered pair of translation lengths $\ell(X),\ell(Y)$ on the Bruhat-Tits tree. Termination is proved by showing that an infinite decreasing sequence of positive integer length pairs would force the limit of the chosen replacement lengths to be non-positive, a contradiction. The exact criterion for when to stop is the corrected axis-overlap formula of Proposition 3.5: it tells whether the axes of $X$ and $Y$ are disjoint, overlap by less than $\min\{\ell(X),\ell(Y)\}$, or overlap by at least that amount, and the inequality used by the algorithm is exactly the condition under which the Ping Pong domains exist.
What would settle it
Take any pair of hyperbolic isometries $A,B$ of a locally finite simplicial tree with $\Delta(A,B) = \min\{\ell(A),\ell(B)\}$ and with both $AB$ and $A^{-1}B$ acting without inversions, and compute $\min\{\ell(AB),\ell(A^{-1}B)\}$ directly from the tree metric; if the value falls outside the four cases listed in Proposition A.1, or if the asserted midpoint $m$ is the midpoint of an edge rather than a vertex, then the corrected formula, and with it the algorithm's stopping rule, would be false.
Extended reading notes
Core claim
The central claim is Theorem 4.2: Algorithm 4.1 terminates in finitely many steps and outputs the correct answer. Given $A,B\in \mathrm{SL}_2(K)$, the algorithm computes translation lengths on the Bruhat-Tits tree through $\ell(X) = -2\min\{0, v(\mathrm{tr}\,X)\}$, then repeatedly replaces the generator pair by $(X,Y)$, $(Y,X)$, $(X^{-1},Y)$, or a pair using $XY$ or $X^{-1}Y$ so that the smaller length never increases. If it meets an elliptic element it returns false; if it reaches a pair with $\lvert\ell(X)-\ell(Y)\rvert < \min\{\ell(XY),\ell(X^{-1}Y)\}$, Corollary 3.6 says the axes of $X$ and $Y$ are disjoint or overlap by less than the smaller translation length, so the Ping Pong Lemma applies and the subgroup is discrete and free of rank two. The correctness proof rests on a complete case analysis of axis overlap for hyperbolic isometries of simplicial trees; the paper's appendix supplies the missing case in the known R-tree version of that analysis.
Load-bearing premise
The proof of the axis-overlap classification for simplicial trees depends on an unstated check that a certain midpoint $m$ is a vertex rather than the midpoint of an edge; if that check fails, the case analysis behind the algorithm's stopping rule has a hole.
Editorial extensions
If this is right
- For any non-archimedean local field $K$, the two-generator discreteness-and-freeness problem for $\mathrm{SL}_2(K)$ is now a finite, implementable decision procedure rather than a case-by-case search.
- If the algorithm returns true, it also outputs generators satisfying the Ping Pong Lemma, so a witness for freeness and discreteness is part of the output.
- The same algorithm applies to two-generated subgroups of the isometry group of any locally finite simplicial tree, provided translation lengths can be computed.
- For the discrete free two-generated subgroups the method recognizes, the constructive membership problem is solvable: given any element of the overgroup, the algorithm either writes it as a word in the generators or proves it is not in the subgroup.
- The corrected translation-length formula in the appendix repairs a gap in an existing R-tree result, so any later construction relying on that formula inherits the extra case.
Reading between the lines
- The paper does not claim it, but the corrected four-case formula for $\ell(\gamma\delta)$ should apply to any action on an $\mathbb{R}$-tree or $\Lambda$-tree where translation lengths are computable, not only to simplicial trees and $\mathrm{SL}_2(K)$.
- A natural next question the paper leaves open is the complexity of the procedure: it proves finite termination, but the number of iterations is controlled only by a decreasing pair of positive integers, and no explicit worst-case bound in terms of the initial entry valuations is given.
- Because the algorithm's arithmetic only needs finitely many $\pi$-adic coefficients of the matrices at each iteration, the same method could plausibly be turned into a certified computation over $\mathbb{Q}_p$ with rigorous precision tracking, though the paper only sketches the truncation scheme.
- One could test the method's boundary by trying to extend it to rank-three or higher subgroups of $\mathrm{SL}_2(K)$: the two-generator Ping Pong certificate would need a more complicated axis-geometry analysis, and the paper does not address that case.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript presents Algorithm 4.1, which takes two elements A,B of SL_2(K) over a non-archimedean local field K and decides whether the subgroup they generate is discrete and free of rank two. The method uses the action of SL_2(K) on the Bruhat-Tits tree, the Ping Pong Lemma, and Nielsen transformations that reduce translation lengths. The termination proof uses a decreasing sequence of positive integer translation-length pairs together with the translation-length formula of Proposition 3.5, and the correctness proof uses Corollary 3.6. Section 5 generalizes the algorithm to isometry groups of locally finite simplicial trees and gives an application to the constructive membership problem. The appendix, joint with F. Paulin, supplies an erratum to Paulin's 1989 Proposition 1.6, including a corrected statement and full proof for R-trees.
Significance. If the results hold, this is a valuable contribution: it provides the first practical decision procedure for discreteness and freeness of two-generator subgroups of SL_2(K) over non-archimedean local fields, directly analogous to known algorithms over the reals. The paper is largely self-contained, gives a detailed termination argument, and includes nontrivial examples demonstrating that the algorithm can require arbitrarily many iterations. A notable strength is the joint erratum correcting a genuine error in a published paper by Paulin, with the missing case identified by examples and proved in full in the appendix. The generalization to locally finite tree isometry groups and the constructive membership application broaden the scope beyond the SL_2(K) setting.
major comments (2)
- [Section 3, Proposition 3.5] The proof of the simplicial-tree version of Proposition 3.5 is incomplete. The text asserts that 'one can check that the assumption that both AB and A^{-1}B act without inversions is sufficient to ensure that this midpoint m is indeed a vertex', but no check is provided, and Theorem 4.2 uses Proposition 3.5 at step (5), making this gap load-bearing. The missing argument is short: if m were the midpoint of an edge e fixed by the relevant product, then that product would map e to itself; since it acts without inversion, it would fix both endpoints of e and hence be elliptic, forcing the corresponding translation length to be 0. Please add this argument explicitly rather than leaving it as an assertion.
- [Section 4, Theorem 4.2] In the termination proof, the sentence 'for each pair (X_n,Y_n) of generators, we are in either case (2)(ii) or the first subcase of (2)(iii) of Proposition 3.5' is asserted without justification. The reader must supply the reasoning that if the axes of X_n and Y_n were disjoint or intersected with Delta < min{l(X_n),l(Y_n)}, then Corollary 3.6 would imply m > l(Y_n)-l(X_n), contradicting that the algorithm has reached step (5); the second subcase of (2)(iii) would give m=0 and termination at step (4). Adding a sentence explaining this point would make the termination argument self-contained.
minor comments (4)
- [Title/Abstract] The title contains a typographical error: 'TWO-GENERA TED' should read 'TWO-GENERATED'.
- [Section 4, implementation paragraph] The discussion of storing coefficients up to M = -r min{0,v(a),...,v(h)} is slightly ambiguous; it would be clearer to say 'up to the coefficient of pi^M' and to note explicitly that M is a non-positive integer (so truncation at pi^0 suffices when all valuations are nonnegative).
- [Section 3, Corollary 3.6] The proof of Corollary 3.6 is concise, but the final equivalence 'if and only if the axes either do not intersect, or intersect along a path of length 0 <= Delta(A,B) < min{l(A),l(B)}' is stated without spelling out that it follows by contrapositive from the remaining cases of Proposition 3.5; a short indication would improve readability.
- [Section 5, Algorithm 5.2] For generality, Algorithm 5.2 should explicitly state that the tree has been subdivided, if necessary, so that the no-inversion hypotheses of Proposition 3.5 hold for all products; the current text mentions this possibility only before Proposition 5.1.
Circularity Check
No significant circularity: the decision procedure is proved from translation-length geometry and the Ping Pong Lemma, not from the discreteness/freeness conclusion.
full rationale
The paper's central claim, Theorem 4.2, is not circular. Algorithm 4.1 computes translation lengths from traces via Proposition 2.3, and its false branches are justified by Proposition 3.1 and Corollary 3.2, which use only the Cayley-Hamilton theorem and compactness of the local field. The true branch is justified by Corollary 3.6: the inequality m > l(Y) - l(X) is shown, using the product-length classification in Proposition 3.5, to be equivalent to the Ping-Pong hypotheses of Proposition 3.4, and Proposition 3.4 explicitly constructs the disjoint closed sets required by the Ping Pong Lemma. Termination is an infinite-descent argument on positive integer translation-length pairs, using only the length relations from Proposition 3.5. Proposition 3.5 is not merely cited: its R-tree version is proved in Appendix A.1 as Proposition A.1, and the paper explicitly corrects the previously published formula in [16, Proposition 1.6] that had a missing case. The only manuscript-flagged weakness is the sentence in the proof of Proposition 3.5 that 'one can check' the no-inversion hypotheses force the midpoint m to be a vertex in the simplicial case. This is a compressed or omitted verification, not a circular reduction: it does not assume discreteness or freeness, and it concerns an independent geometric fact about fixed midpoints and translation lengths. There are no fitted parameters, no quantity named a prediction that is actually an input, and no load-bearing self-citation. The correction to [16] is supported by an explicit proof and by the independent Alperin-Bass reference, so the citation is not the argument.
Assumptions & free parameters
assumptions (5)
- standard math SL_2(K) acts by isometries and without inversions on the Bruhat-Tits tree (Serre, [19])
- standard math Translation length formula l(A) = -2 min{0, v(tr(A))} for A in SL_2(K) (Morgan-Shalen, [14, Prop. II.3.15])
- standard math Ping Pong Lemma for metrisable topological groups (Lemma 3.3, proved in paper)
- domain assumption For a proper metric space X, Isom(X) with the topology of pointwise convergence is a metrisable topological group (Cornulier-de la Harpe, [6])
- standard math Every hyperbolic isometry of a simplicial tree has an axis, and d(q,gq) = l(g)+2k for vertices q at distance k from the axis (Serre, [19], Prop. 2.1)
Cite this review
Pith. "Pith review of Discrete and free two-generated subgroups of ${\rm SL_2}$ over non-archimedean local fields." pith.science (2026). https://pith.science/paper/PFVKN3TC
@misc{pith2026190811114,
author = {Pith},
title = {Pith review of: Discrete and free two-generated subgroups of $\rm SL_2$ over non-archimedean local fields},
year = {2026},
howpublished = {\url{https://pith.science/paper/PFVKN3TC}},
note = {Machine review of arXiv:1908.11114}
}
abstract
We present a practical algorithm which, given a non-archimedean local field $K$ and any two elements $A,B\in {\rm SL_2}(K)$, determines after finitely many steps whether or not the subgroup $\langle A, B \rangle\le {\rm SL_2}(K)$ is discrete and free of rank two. This makes use of the Ping Pong Lemma applied to the action of ${\rm SL_2}(K)$ by isometries on its Bruhat-Tits tree. The algorithm itself can also be used for two-generated subgroups of the isometry group of any locally finite simplicial tree, and has applications to the constructive membership problem. In an appendix joint with Fr\'ed\'eric Paulin, we give an erratum to his 1989 paper `The Gromov topology on $\mathbb{R}$-trees', which details some translation length formulae that are fundamental to the algorithm.
Figures
Reference graph
Works this paper leans on
-
[1]
R. Alperin and H. Bass. Length functions of group actions on Λ-trees. In Combinatorial Group Theory and Topology, volume 111 of Ann. of Math. Stud. , pages 265–378. Princeton Univ. Press, 1987
work page 1987
-
[2]
S. Alvarez, D. Filimonov, V. Kleptsyn, D. Malicet, C. Meni˜ no Cot´ on, A. Navas, and M. Triestino. Groups with infinitely many ends acting analytically on the circle. J. Topol., 12(4):1315–1367, 2019
work page 2019
-
[3]
A. F. Beardon. Pell’s equation and two generator free M¨ obius groups. Bull. London Math. Soc., 25(6):527–532, 1993
work page 1993
-
[4]
B. H. Bowditch. Markoff triples and quasi-Fuchsian groups. Proc. London Math. Soc. , 77(3):697–736, 1998
work page 1998
-
[5]
J. W. S. Cassels. Local Fields. Cambridge University Press, 1986
work page 1986
-
[6]
Y. Cornulier and P. de la Harpe. Metric Geometry of Locally Compact Groups . European Mathematical Society, 2016
work page 2016
-
[7]
M. Culler and J. W. Morgan. Group actions on R-trees. Proc. London Math. Soc., 55(3):571– 604, 1987
work page 1987
-
[8]
M. Culler and K. Vogtmann. The boundary of outer space in rank two. In Arboreal Group Theory, volume 19 of Math. Sci. Res. Inst. Publ. , pages 189–230. Springer, 1991
work page 1991
Show all 20 references
-
[9]
B. Eick, M. Kirschmer, and C. Leedham-Green. The constructive membership problem for discrete free subgroups of rank 2 of SL 2(R). LMS J. Comput. Math. , 17(1):345–359, 2014
2014
-
[10]
Gaboriau and G
D. Gaboriau and G. Levitt. The rank of actions on R-trees. Ann. Sci. ´Ecole Norm. Sup. (4) , 28(5):549–570, 1995
1995
-
[11]
J. Gilman. Two-generator discrete subgroups of PSL 2(R). Mem. Amer. Math. Soc., 117(561), 1995
1995
-
[12]
Guirardel and G
V. Guirardel and G. Levitt. Deformation spaces of trees. Groups Geom. Dyn., 1(2):135–181, 2007
2007
-
[13]
R. C. Lyndon and J. L. Ullman. Groups generated by two parabolic linear fractional trans- formations. Canadian J. Math. , 21:1388–1403, 1969
1969
-
[14]
J. W. Morgan and P. B. Shalen. Valuations, trees, and degenerations of hyperbolic structures. I. Ann. of Math. (2) , 120(3):401–476, 1984
1984
-
[15]
M. Newman. Pairs of matrices generating discrete free groups and free products. Michigan Math. J., 15:155–160, 1968
1968
-
[16]
F. Paulin. The Gromov topology on R-trees. Topology Appl., 32(3):197–221, 1989
1989
-
[17]
Purzitsky
N. Purzitsky. Two-generator discrete free products. Math. Z., 126:209–223, 1972
1972
-
[18]
Rosenberger
G. Rosenberger. Fuchssche Gruppen, die freies Produkt zweier zyklischer Gruppen sind, und die Gleichung x2 +y2 +z2 =xyz. Math. Ann., 199:213–227, 1972
1972
-
[19]
J-P. Serre. Trees. Springer, 1980. Translated by John Stillwell
1980
-
[20]
Urba´ nski and L
M. Urba´ nski and L. Zamboni. On free actions on Λ-trees. Math. Proc. Cambridge Philos. Soc., 113(3):535–542, 1993. M. J. Conder, Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge,...
1993
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