{"id":"1b2593da-b372-4059-90b8-f083a43ca5b3","arxiv_id":"2505.13639","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every lattice in SL_3(k) over a nonarchimedean local field k contains an undistorted subgroup isomorphic to Z^2 * Z, yielding new discrete subgroups not virtually isomorphic to lattices.","lead":"This paper proves that every lattice in SL(3) over a nonarchimedean local field contains a free product subgroup of the form Z^2 * Z that is undistorted. Such subgroups give the first known examples of discrete subgroups of nonarchimedean Lie groups that are not virtually lattices.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1 omits the valuation case analysis that justifies the key disjointness assertion for the ping-pong sets.","rationale":"The Reader correctly identified the unproved valuation assertion as the load-bearing step in the proof of Theorem 1. Without it, the sets used for the ping-pong argument are not shown to be disjoint, and the free-product decomposition does not follow. The paper's one-line justification 'The latter is true since either σ = ∞ or val_π(σ) ≠ 1' is not a proof. However, a careful inspection of the valuations shows that the assertion is in fact correct under the hypotheses: the diagonal forms of the generators force r = val_π(α) = val_π(β) to be a positive multiple of 3, and a sign case analysis gives val_π(σ) never equal to 1. Thus the gap is an omitted argument, not a counterexample. The rest of the proof—the structure of the ping-pong argument, the use of Proposition 6 for undistortion, and the finite-index passage—is coherent, and the cited external results are used in a plausible way. The Reader's CONDITIONAL verdict is therefore appropriate: the theorem is likely true and the gap appears fillable, but the manuscript as written is incomplete at its central point. I did not find a more serious load-bearing concern; the potential issue in Remark 4 about g not fixing any coordinate flag is secondary to the main theorem and does not affect the proof of Theorem 1. Hence the verdict should remain unchanged.","tokens_in":6913,"tokens_out":18321,"duration_ms":173440,"concrete_test":"Perform the omitted valuation case analysis explicitly. Let r = val_π(α) = val_π(β). Use that 1+λ_i, 1+μ_i are units and that r ≥ 3 is a multiple of 3, which follows from |a1|=|b2|<1, |a2|=|a3|=|b1|=|b3| and det=1. Split into the six cases (m,n) with m,n > 0; m,n < 0; m>0,n<0; m<0,n>0; m=0,n≠0; and n=0,m≠0. In each case compute val_π of numerator and denominator of σ, showing val_π(σ) is either ∞, 0, a nonzero multiple of r, or a value clearly different from 1. If this computation reproduces the claimed bound, the gap is fillable and the theorem holds; if some case yields val_π(σ)=1, the disjointness claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1, the claim that V_x ∩ γU = ∅ is reduced to showing that the slope σ = [β^n(1+λ2) − (1+μ2)] / [α^m(1+λ1) − (1+μ1)] is not in π + πm. The proof then says, without further argument, that this holds because either σ = ∞ or val_π(σ) ≠ 1. This is the single step that makes the ping-pong sets disjoint: if some choice of signs of m,n and elements of 1+πm gave val_π(σ) = 1, then a line through x of slope in π+πm would meet γU, destroying the free-product decomposition. The text provides no case analysis for the possible signs of m and n, nor any valuation computation. The assertion is load-bearing and, as written, unproved. A short argument shows it is likely true: writing r = val_π(α) = val_π(β), the determinant condition and |a1| = |b2| < 1, |a2| = |a3| = |b1| = |b3| force r to be a positive multiple of 3, and a sign split on (m,n) yields val_π(σ) ∈ {∞, 0, ±kr} or a value with opposite sign to a positive term, never 1. But this argument is absent from the paper, leaving a genuine gap in the proof of the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves (or aims to prove) that every lattice Λ in SL3(k), for a nonarchimedean local field k, contains an undistorted subgroup isomorphic to the free product Z^2 * Z. The proof uses a ping-pong argument on the Furstenberg boundary of SL3(k): it exhibits a finite-index subgroup Δ of a given Z^2 subgroup of Λ, a diagonal affine action on a chart, and a compact set U = (1+πm)^2 such that a certain set V_x of points and lines through x is disjoint from γU for every nontrivial γ∈Δ. A regular element g∈Λ with attracting and repelling flags in a related open set then gives the free product decomposition Δ∗⟨g⟩. The paper also contains a quantitative ping-pong lemma (Proposition 6) used to prove undistortedness, and remarks explaining Zariski density and why the constructed group is not virtually isomorphic to a lattice.","tokens_in":7222,"tokens_out":14421,"duration_ms":141783,"significance":"If the main theorem is correct, the paper provides the first known finitely generated discrete subgroups of nonarchimedean Lie groups that are not virtually isomorphic to lattices, in contrast to the open problem for SL3(Z). The overall strategy is natural and the paper is well structured: Proposition 6 is a clean, explicit ping-pong lemma with a convincing proof, and the reduction to the affine chart is geometrically reasonable. The construction is new and does not appear circular: the result is derived from standard external theorems about arithmetic lattices, tori, and Bruhat–Tits buildings. The central obstruction to accepting the proof as written is a single unproved valuation assertion, which is load-bearing for the ping-pong disjointness.","major_comments":[{"comment":"The proof of the disjointness claim ends with the sentence 'The latter is true since either σ = ∞ or val_π(σ) ≠ 1', but no argument is supplied for this valuation assertion. This assertion is exactly what makes the ping-pong sets disjoint: if a line through x had slope in π+πm and met γU, the claimed free product decomposition would not follow. The text provides no case analysis for the signs of m and n, no computation of val_π(α) and val_π(β), and no treatment of how the factors 1+λ_i and 1+μ_i affect possible cancellations. A short case analysis may well prove the assertion, but as written it is a genuine gap in the proof of Theorem 1.","section":"Proof of Theorem 1, final sentence of the claim"}],"minor_comments":[{"comment":"The abstract as quoted at the beginning of the submission and the abstract printed in the body of the paper make different novelty claims: the former says 'not virtually isomorphic to lattices', while the latter says 'not virtually free'. These are not equivalent statements, and the paper should state its intended claim consistently, ideally matching the formulation supported by Remark 5.","section":"Abstract"},{"comment":"The sentence 'It is clear that U ∩ γU = ∅' is asserted without explanation. Since γ is an arbitrary nontrivial element of Δ and may have mixed expansion and contraction in the affine coordinates, a one-line valuation justification would improve the exposition.","section":"Proof of Theorem 1, first paragraph of the claim"},{"comment":"The displayed text contains many instances of the artifact '/integerdivide' where a set-difference or quotient symbol is clearly intended. These should be corrected in the final version so that formulas such as P(k^3) \\ V_{x±} are readable.","section":"Throughout"},{"comment":"The reduction 'Note that β1^n, α2^m ∈ 1+πm' is stated without proof. It is plausible from the preceding normalization using p(q−1)-powers, but the authors should spell out why α2 and β1 become elements of 1+πm after that normalization.","section":"Proof of Theorem 1, definition of σ"}],"recommendation":"major_revision","confidential_remarks":"The identified gap in the valuation assertion is likely repairable within the scope of the paper, but it is central to the proof and should not be left as an unproved sentence. I also recommend harmonizing the two versions of the abstract. If the authors supply the missing valuation case analysis, the paper would be a solid contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe one thing to know: this paper is the first to produce finitely generated discrete subgroups of nonarchimedean Lie groups that are not virtually isomorphic to lattices. It proves every lattice in SL_3(k) contains an undistorted free product Z^2 * Z, which is a real advance and a stark contrast with the open SL_3(Z) question.\n\nWhat is new: the ping-pong construction on the projective plane over k, with separating sets defined using valuation balls. The choice of U = (1+πm)^2 and slopes in π+πm is clever, and the reduction to a free product is clean. Proposition 6, the quasi-isometric embedding lemma, is standard and carefully proved. The use of Margulis arithmeticity and Prasad–Rapinchuk is appropriate. The paper is well-written and the claims are precise.\n\nThe soft spot: in the proof of Theorem 1, the claim that V_x ∩ γU = ∅ hinges on the final sentence: the slope σ satisfies either σ = ∞ or val_π(σ) ≠ 1. This is asserted without proof. It is the only step that makes the ping-pong sets disjoint, so it is load-bearing. The stress-test sketch suggests a case analysis on signs and valuations would confirm it, so the gap looks fillable, but the paper does not contain that analysis. A referee should ask for it.\n\nMinor issue: the PDF abstract says the subgroups are 'not virtually free', while the stated abstract and Remark 5 claim 'not virtually isomorphic to lattices'. These are different claims; the body supports the latter. The authors should align the wording.\n\nBottom line: for people working on p-adic discrete subgroups, this deserves a serious referee. I would send it out; the gap is localized and likely repairable. If it fills, this is a solid contribution.","headline":"First discrete non-lattice subgroups in nonarchimedean SL_3, built by a clever ping-pong argument that has one unproved but likely fillable valuation bound.","tokens_in":7713,"tokens_out":2463,"would_cite":true,"duration_ms":23423,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E40","20F65","20E06","20G25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every lattice in SL(3) over a nonarchimedean field contains an undistorted Z^2 * Z subgroup.","keywords":["lattices in Lie groups","nonarchimedean local fields","free products","undistorted subgroups","ping-pong lemma","projective flags","algebraically dense discrete subgroups","function fields"],"falsifier":"Compute, for a concrete nonarchimedean field such as $\\mathbb{Q}_p$ or $\\mathbb{F}_q((t))$, the slope $\\sigma$ appearing in the last paragraph of the proof for a specific diagonal $\\mathbb{Z}^2$ subgroup and specific $x,y\\in U$; if any choice gives $\\mathrm{val}_\\pi(\\sigma)=1$, then $V_x\\cap \\gamma U$ is nonempty and the free product decomposition claimed in Theorem 1 would fail for that configuration.","tokens_in":6748,"feed_emoji":"🏓","tokens_out":16404,"duration_ms":144145,"temperature":0.7,"pith_summary":"The paper proves that every lattice in $\\mathrm{SL}_3(k)$ — a discrete subgroup whose quotient has finite volume — contains an undistorted subgroup isomorphic to the free product $\\mathbb{Z}^2 * \\mathbb{Z}$ (the group of alternating words in the two factors), for every nonarchimedean local field $k$ such as a p-adic field. The proof sets up a ping-pong action on the projective plane over $k$: a finite-index copy of $\\mathbb{Z}^2$ is conjugated into diagonal form, and a carefully chosen regular element sends the complement of a family of lines into a small neighborhood, forcing the free product decomposition. Undistorted means the subgroup sits inside the ambient lattice in a way that preserves metric growth. These would be the first known finitely generated discrete subgroups of nonarchimedean Lie groups that are not virtually isomorphic to any lattice, in contrast to $\\mathrm{SL}_3(\\mathbb{Z})$, where the existence of a $\\mathbb{Z}^2 * \\mathbb{Z}$ subgroup remains open.","feed_headline":"Every nonarchimedean SL(3) lattice hides an undistorted free product","feed_subtitle":"The new subgroups are discrete, infinite-covolume, and not virtually isomorphic to any lattice.","key_machinery":"The load-bearing mechanism is a two-set ping-pong on the projective plane and its flag space. For $U = (1+\\pi\\mathfrak{m})^2$ inside the affine chart, and for each $x\\in U$, let $V_x$ be the union of $U$ with every affine line through $x$ whose slope lies in $\\pi+\\pi\\mathfrak{m}$. The paper's key combinatorial claim is that $V_x \\cap \\gamma U = \\varnothing$ for every nontrivial $\\gamma$ in the diagonal $\\mathbb{Z}^2$ subgroup; this disjointness is what turns the action into a free product. A regular element $g$ whose attracting and repelling flags both lie in the associated open set $W$ then sends the complement of the two regions $V_{x_\\pm}$ into $U$, so the ping-pong lemma applies. To control distortion, the paper uses an operator-norm pinning estimate (Proposition 6): the ping-pong inequalities are converted into a lower bound of the form $\\|g\\|\\|g^{-1}\\| \\geq e^{\\alpha |g| - C}$, which is the quantitative meaning of undistorted in this setting.","core_discovery":"At its heart the paper establishes Theorem 1: for every lattice $\\Lambda < \\mathrm{SL}_3(k)$ and every $\\mathbb{Z}^2$ subgroup $\\Delta' \\subset \\Lambda$, there is a finite-index subgroup $\\Delta \\subset \\Delta'$ and an infinite-order element $g \\in \\Lambda$ such that $\\langle \\Delta, g\\rangle$ is undistorted and decomposes as the free product $\\Delta * \\langle g\\rangle$. The desired free factor $\\Delta$ is a diagonal $\\mathbb{Z}^2$ acting on the affine chart $\\{Z\\neq 0\\}$ of $\\mathbb{P}(k^3)$, with eigenvalues of norm less than $1$ in the relevant coordinates. The proof identifies an open set $W$ of projective flags $(x,L)$ in which $x$ lies near $(1,1)$ and the line $L$ has slope in $\\pi + \\pi\\mathfrak{m}$; a regular element $h$ with attracting and repelling flags in $W$ then acts as the second ping-pong player after passing to a suitable power $g = h^{N_0}$. The paper also shows directly that the resulting subgroup is algebraically dense and infinite covolume, and argues that no group of the form $\\mathbb{Z}^2 * \\mathbb{Z}$ can be a lattice in any local field, so the construction really produces groups of a new discreteness type.","pith_inferences":["The final step of the disjointness proof asserts, without proof, that the slope $\\sigma$ of the line joining $\\gamma x$ to $y$ always satisfies $\\sigma=\\infty$ or $\\mathrm{val}_\\pi(\\sigma)\\neq 1$; verifying this valuation bound, or finding a counterexample, is the natural next check. If the bound fails, the ping-pong sets could overlap and the free-product conclusion would not follow.","The same slope-controlled construction could plausibly extend to other rank-two groups over nonarchimedean fields by replacing the condition 'slope in $\\pi+\\pi\\mathfrak{m}$' with an analogous residue-condition, though the paper does not claim such an extension.","The contrast with the open $\\mathrm{SL}_3(\\mathbb{Z})$ case suggests that the obstruction is boundary geometry rather than group theory: ultrametric valuation constraints can guarantee disjointness in a way that archimedean absolute values cannot."],"forward_implications":["Corollary 3 follows: the lattice $\\mathrm{SL}_3(\\mathbb{F}_q[t])$ in $\\mathrm{SL}_3(\\mathbb{F}_q((1/t)))$ contains a subgroup isomorphic to $\\mathbb{Z}^2 * \\mathbb{Z}$.","The subgroup is algebraically dense and of infinite covolume, so it provides the first examples of finitely generated algebraically dense infinite-covolume discrete subgroups of a nonarchimedean almost-simple group that are not virtually free.","Because $\\mathbb{Z}^2 * \\mathbb{Z}$ cannot embed as a lattice in the $k$-points of any $k$-group, the theorem produces finitely generated discrete subgroups of nonarchimedean Lie groups that are not virtually isomorphic to any lattice.","For $\\mathrm{SL}_3(\\mathbb{Z})$ the existence of a $\\mathbb{Z}^2 * \\mathbb{Z}$ subgroup remains open; the paper observes that a hypothetical such subgroup would act minimally on the flag boundary, so the ping-pong method used here cannot transfer to that case."],"supporting_citations":[{"why":"Supplies the building fact that a discrete Z^2 subgroup preserves an apartment, allowing conjugation into the diagonal form used in the affine chart.","marker":"[3]"},{"why":"Provides the nonarchimedean definition of the eigenvalue-counting map used in the quasi-isometric embedding criterion behind Proposition 6.","marker":"[4]"},{"why":"The arithmeticity theorem for lattices, invoked to guarantee that every lattice contains a Z^2 subgroup and again in the non-lattice argument.","marker":"[9]"},{"why":"Supplies the existence of irreducible tori (and hence Z^2 subgroups) as well as the regular element whose attracting and repelling flags lie in the needed open set.","marker":"[11]"},{"why":"Lattice-envelope theorem used in Remark 5 to rule out the possibility that Z^2*Z itself is a lattice.","marker":"[1]"},{"why":"Tree-lattice result that nonarchimedean rank-one lattices are not finitely generated, used to close the contradiction in Remark 5.","marker":"[2]"},{"why":"Rigidity theorem for lattices in semisimple groups, used in Remark 5 to exclude archimedean rank-one lattice embeddings of Z^2*Z.","marker":"[10]"}],"fun_headline_variants":["Nonarchimedean SL(3) lattices hide a Z^2*Z subgroup","First non-lattice discrete subgroups from nonarchimedean ping-pong","Undistorted Z^2*Z in every SL(3) lattice over local fields","Ping-pong gives new discreteness type in nonarchimedean Lie groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction depends on the final, unproved assertion in the proof of Theorem 1 that the slope $\\sigma$ of a certain line always satisfies $\\sigma=\\infty$ or $\\mathrm{val}_\\pi(\\sigma)\\neq 1$, because that valuation bound is what keeps the two ping-pong sets disjoint.","fun_headline_variants_meta":{"raw":{"variants":["Nonarchimedean SL(3) lattices hide a Z^2*Z subgroup","First non-lattice discrete subgroups from nonarchimedean ping-pong","Undistorted Z^2*Z in every SL(3) lattice over local fields","Ping-pong gives new discreteness type in nonarchimedean Lie groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000479,"raw_usage":{"total_tokens":2367,"prompt_tokens":938,"completion_tokens":1429,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":1337}},"tokens_in":554,"tokens_out":1429,"duration_ms":10653,"temperature":1.0,"reasoning_tokens":1337,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:14:40.555457+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a concrete nonarchimedean field such as $\\mathbb{Q}_p$ or $\\mathbb{F}_q((t))$, the slope $\\sigma$ appearing in the last paragraph of the proof for a specific diagonal $\\mathbb{Z}^2$ subgroup and specific $x,y\\in U$; if any choice gives $\\mathrm{val}_\\pi(\\sigma)=1$, then $V_x\\cap \\gamma U$ is nonempty and the free product decomposition claimed in Theorem 1 would fail for that configuration.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the building fact that a discrete Z^2 subgroup preserves an apartment, allowing conjugation into the diagonal form used in the affine chart."},{"cited_title":"Bruhat and J","cited_arxiv_id":null,"evidence_quote":"Provides the nonarchimedean definition of the eigenvalue-counting map used in the quasi-isometric embedding criterion behind Proposition 6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The arithmeticity theorem for lattices, invoked to guarantee that every lattice contains a Z^2 subgroup and again in the non-lattice argument."},{"cited_title":"Prasad and A","cited_arxiv_id":null,"evidence_quote":"Supplies the existence of irreducible tori (and hence Z^2 subgroups) as well as the regular element whose attracting and repelling flags lie in the needed open set."},{"cited_title":"Bader, A","cited_arxiv_id":null,"evidence_quote":"Lattice-envelope theorem used in Remark 5 to rule out the possibility that Z^2*Z itself is a lattice."},{"cited_title":"Bass and A","cited_arxiv_id":null,"evidence_quote":"Tree-lattice result that nonarchimedean rank-one lattices are not finitely generated, used to close the contradiction in Remark 5."},{"cited_title":"Prasad , Discrete subgroups isomorphic to lattices in semisimple Li e groups , Amer","cited_arxiv_id":null,"evidence_quote":"Rigidity theorem for lattices in semisimple groups, used in Remark 5 to exclude archimedean rank-one lattice embeddings of Z^2*Z."}],"review_version":1}