{"id":"ecf080f0-7f9a-44f6-bf35-816f4a5f0493","arxiv_id":"1908.11114","paper_version":3,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A terminating algorithm decides, for any two elements of SL_2 over a non-archimedean local field, whether the subgroup they generate is discrete and free of rank two.","lead":"This paper gives a step-by-step algorithm that decides whether two matrices over a p-adic-like field generate a discrete free group, by measuring how they move points on a tree. It also corrects a 1989 formula about tree translations that the algorithm depends on.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader correctly identifies the midpoint check in Proposition 3.5 as the weakest point in the paper, and I agree that it is the only place where the proof is not fully self-contained. However, the concern does not land as a correctness objection. If the midpoint m fixed by the relevant product is an edge-midpoint, then the product fixes that edge setwise; since it acts without inversion, it fixes both endpoints and is elliptic. Hence the translation length is 0 exactly as the formula in Proposition 3.5(2)(iii) predicts. The termination proof in Theorem 4.2 is also sound: it relies on non-increasing positive integer translation lengths and the replacement relation x_{n+1} + y_{n+1} = y_n - k_n; the only slight looseness is that the sequence k_n may not converge, but a subsequence argument yields the same contradiction. Independent support includes the joint appendix with Paulin, which corrects and fully proves the R-tree version of the formula. Therefore the central claim remains well supported and no verdict change is warranted.","tokens_in":14903,"tokens_out":22457,"duration_ms":231280,"concrete_test":"Write out the simplicial-tree version of the two cases in Prop. A.1(2)(ii) where gamma delta fixes a midpoint m (the second and third bullets), and check the following: if m lies in the interior of an edge e, then gamma delta maps e to itself; because gamma delta acts without inversion on the simplicial tree, it fixes both endpoints of e, so l(gamma delta) = 0. Repeat this for both AB and A^{-1}B and compare with the formula in Proposition 3.5(2)(iii). If this verification succeeds, the only unproved step in the proof of Theorem 4.2 is closed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing objection identified. The closest candidate is the transfer from the R-tree proposition to simplicial trees in Proposition 3.5. The proof says only that the no-inversion hypotheses on AB and A^{-1}B allow 'one to check' that the midpoint m fixed by the relevant product is a vertex. Theorem 4.2 uses Proposition 3.5 at step (5), so an unfixable error here would hurt the central claim. But the check is easily supplied: if m is the midpoint of an edge e and the product fixes m, then it maps e to itself; since the product acts without inversion, it fixes both endpoints of e and is elliptic, so the translation length is 0, matching the formula in Proposition 3.5(2)(iii). A one-paragraph addition closes the gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14991,"tokens_out":6436,"duration_ms":62109,"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":[{"comment":"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":"Section 3, Proposition 3.5"},{"comment":"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.","section":"Section 4, Theorem 4.2"}],"minor_comments":[{"comment":"The title contains a typographical error: 'TWO-GENERA TED' should read 'TWO-GENERATED'.","section":"Title/Abstract"},{"comment":"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":"Section 4, implementation paragraph"},{"comment":"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":"Section 3, Corollary 3.6"},{"comment":"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.","section":"Section 5, Algorithm 5.2"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a solid contribution that corrects a published error and provides a useful algorithm. The two major comments identify missing arguments that are short to supply, so I do not see a need for a full major revision. The paper fits the scope of a general mathematics journal with a group-theoretic audience."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe thing to know: this paper delivers what the abstract promises—a terminating algorithm that decides whether a two-generated subgroup of SL2 over a non-archimedean local field is discrete and free of rank two. That is new; the real case was handled by Eick–Kirschmer–Leedham-Green, but the non-archimedean case was open. The algorithm is clean: use translation lengths on the Bruhat-Tits tree, run trace-minimizing Nielsen transformations, and either hit an elliptic element or find a pair satisfying the Ping Pong hypotheses.\n\nThe paper does several things well. The erratum to Paulin's Proposition 1.6 is real and important. The counterexample in SL2(Q7) is explicit, and the joint appendix with Paulin gives a full corrected proof in the R-tree setting. The translation length formulae in Proposition 3.5 are central, and the extra case (2)(iii) is genuinely missing from the 1989 paper. The termination proof for Algorithm 4.1 is sound, though it argues by limits when a direct sum-decrease argument would be simpler. The generalization to locally finite simplicial trees and the constructive membership algorithm are natural and well explained.\n\nWhere are the soft spots? Proposition 3.5 is the one place I'd want a patch. The proof of the simplicial-tree version says 'one can check' that the midpoint is a vertex when AB and A^{-1}B act without inversions. That check is not written out. It is not hard—if the midpoint were in an edge, the product would fix the edge and hence invert it, contradicting the no-inversion assumption—but the paper should say that. The appendix proves the R-tree version in detail, so the gap is small and clearly fillable. The implementation discussion about finite-precision storage is a bit hand-wavy but honest; the examples are sufficient to show practicality.\n\nCitation pattern: fine. The self-citation to Paulin is because they are correcting it, which is exactly when self-citation is appropriate. No parameter fitting, no circularity.\n\nVerdict: accept. This deserves a serious referee. The main theorem is true as far as I can tell, the erratum is valuable, and the algorithm is a genuine contribution. I'd bring it to reading group and would cite it.","headline":"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.","tokens_in":15530,"tokens_out":4740,"would_cite":true,"duration_ms":41668,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20E08","20F65","20G25"],"pacs":[],"model":"deepseek-v4-flash","headline":"A finite algorithm decides whether any two matrices over a non-archimedean local field generate a discrete free subgroup of rank two.","keywords":["discrete free subgroups","SL2 over non-archimedean local fields","Bruhat-Tits tree","Ping Pong Lemma","Nielsen transformations","translation length","constructive membership problem","R-tree erratum"],"falsifier":"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.","tokens_in":14689,"feed_emoji":"🌳","tokens_out":10976,"duration_ms":100668,"temperature":0.7,"pith_summary":"The paper's aim is to turn the question \"do these two matrices generate a discrete free group?\" into a finite computation. Over a non-archimedean local field, the answer is found by watching how the matrices move the Bruhat-Tits tree: repeated Nielsen transformations shrink translation lengths until either an elliptic element appears, which rules out discreteness and freeness, or the generators satisfy an inequality that triggers the Ping Pong Lemma and certifies that the group is discrete and free. The same procedure works for two-generated subgroups of the isometry group of any locally finite simplicial tree, and it supplies a constructive membership test for the groups it recognizes. Along the way the paper identifies and corrects a missing case in a standard formula for the translation length of a product of hyperbolic isometries; that correction is what makes the stopping rule exact.","feed_headline":"Finite algorithm decides discrete free pairs in p-adic SL2","feed_subtitle":"Translation-length minimization plus the Ping Pong Lemma settles which two-matrix subgroups are discrete and free.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It supplies the original translation-length formula for products of hyperbolic isometries; the paper finds a missing case in it and corrects it in the appendix, and the corrected formula underpins the algorithm's stopping rule.","marker":"[16]"},{"why":"It provides the template trace-minimizing Nielsen algorithm for $\\mathrm{SL}_2(\\mathbb{R})$ and the constructive membership method that this paper adapts to the tree setting.","marker":"[9]"},{"why":"It supplies the standard theory of groups acting on simplicial trees, including the axis and translation-length facts used in Proposition 3.4 and the Bass-Serre tree for amalgamated products.","marker":"[19]"},{"why":"It gives the trace-to-translation-length formula $\\ell(A) = -2\\min\\{0, v(\\mathrm{tr}\\,A)\\}$ that lets the algorithm compute lengths directly from matrix entries.","marker":"[14]"},{"why":"It supplies the ping-pong and fundamental-domain argument for pairs of hyperbolic isometries of trees that Proposition 3.4 and Algorithm 5.4 rely on.","marker":"[7]"},{"why":"It provides the R-tree length-function facts used in the appendix's corrected proof of the translation-length formula.","marker":"[1]"},{"why":"It supplies the normal-form translation-length fact used to apply the algorithm to amalgamated free products.","marker":"[2]"}],"fun_headline_variants":["Finite algorithm decides free discrete pairs via ping-pong","Ping-pong on Bruhat-Tits tree gives finite decision test","Decide discrete free subgroups in SL2 with finite steps","Tree-based algorithm: finite check for free discrete pairs","Ping-pong lemma powers finite test for free discrete subgroups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Finite algorithm decides free discrete pairs via ping-pong","Ping-pong on Bruhat-Tits tree gives finite decision test","Decide discrete free subgroups in SL2 with finite steps","Tree-based algorithm: finite check for free discrete pairs","Ping-pong lemma powers finite test for free discrete subgroups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000195,"raw_usage":{"total_tokens":1369,"prompt_tokens":966,"completion_tokens":403,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":582,"completion_tokens_details":{"reasoning_tokens":319}},"tokens_in":582,"tokens_out":403,"duration_ms":4319,"temperature":1.0,"reasoning_tokens":319,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:23:25.561672+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the original translation-length formula for products of hyperbolic isometries; the paper finds a missing case in it and corrects it in the appendix, and the corrected formula underpins the algorithm's stopping rule."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the template trace-minimizing Nielsen algorithm for $\\mathrm{SL}_2(\\mathbb{R})$ and the constructive membership method that this paper adapts to the tree setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the standard theory of groups acting on simplicial trees, including the axis and translation-length facts used in Proposition 3.4 and the Bass-Serre tree for amalgamated products."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives the trace-to-translation-length formula $\\ell(A) = -2\\min\\{0, v(\\mathrm{tr}\\,A)\\}$ that lets the algorithm compute lengths directly from matrix entries."},{"cited_title":"Culler and J","cited_arxiv_id":null,"evidence_quote":"It supplies the ping-pong and fundamental-domain argument for pairs of hyperbolic isometries of trees that Proposition 3.4 and Algorithm 5.4 rely on."},{"cited_title":"Alperin and H","cited_arxiv_id":null,"evidence_quote":"It provides the R-tree length-function facts used in the appendix's corrected proof of the translation-length formula."},{"cited_title":"Alvarez, D","cited_arxiv_id":null,"evidence_quote":"It supplies the normal-form translation-length fact used to apply the algorithm to amalgamated free products."}],"review_version":1}