{"id":"ba5fd3e1-af9f-407c-94ff-3cde3b696392","arxiv_id":"2607.00697","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Lattices in simple p-adic groups have only finite C1 actions on compact manifolds of dimension less than the rank; extended to totally disconnected S-arithmetic groups with critical dimension equal to maximal rank of simple factors.","lead":"The paper proves that actions by C1 diffeomorphisms of lattices in simple p-adic groups on compact manifolds are finite when manifold dimension is less than the group rank, and extends the result to S-arithmetic groups. A smart generalist might read it to understand constraints on symmetries and rigidity in group actions on geometric spaces.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Applicability of BFH machinery to p-adic and totally disconnected groups needs explicit hypothesis verification","rationale":"The reader's weakest_assumption is exactly the load-bearing point. The abstract states only that the result 'uses the machinery,' so the full text's value hinges on whether the adaptation is carried out rigorously; without that check the claim cannot be accepted at face value.","tokens_in":1549,"tokens_out":332,"duration_ms":19026,"concrete_test":"In the section invoking the BFH theorem, extract the precise statement cited and the list of its hypotheses; check whether each hypothesis is explicitly confirmed for a p-adic lattice (e.g., SL(3, Q_p) on a 2-manifold) with page references or lemmas. If any hypothesis is only asserted without proof or adaptation, the extension is unsupported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is that C^1 actions of lattices in simple p-adic groups (and extensions to totally disconnected S-arithmetic groups) on compact manifolds are finite when dim(M) < rank (or max rank of factors). This rests entirely on the Brown-Fisher-Hurtado finiteness theorem applying verbatim. BFH was developed for lattices in connected real semisimple Lie groups and uses structure such as Cartan subgroups, real-rank dynamics on the derivative cocycle, and connectedness for rigidity results. The manuscript must verify that these (or suitable replacements) hold for groups like SL(n, Q_p), whose topology is totally disconnected, and that the rank definition carries over without altering the dimension threshold.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that any C^1 action by diffeomorphisms of a lattice in a simple p-adic group on a compact manifold is finite provided the manifold dimension is strictly less than the rank; it extends the result to lattices in totally disconnected S-arithmetic groups, replacing the rank threshold by the maximal rank among the simple factors. The proof is obtained by invoking the Brown-Fisher-Hurtado finiteness theorem.","tokens_in":1683,"tokens_out":293,"duration_ms":12460,"significance":"If the central claim holds, the result would constitute a genuine extension of smooth rigidity phenomena from connected real semisimple Lie groups to p-adic and totally disconnected settings, thereby enlarging the class of groups for which low-dimensional actions are forced to be finite.","major_comments":[{"comment":"The manuscript invokes the Brown-Fisher-Hurtado theorem without an explicit check that its hypotheses are satisfied for lattices in simple p-adic groups. In particular, the original BFH argument relies on the existence of Cartan subgroups, real-rank dynamics on the derivative cocycle, and connectedness properties that are absent or require replacement in the totally disconnected topology of groups such as SL(n, Q_p). Because the entire finiteness statement rests on this applicability, the omission is load-bearing.","section":"Abstract / main argument"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading of the manuscript and for pointing out the need for an explicit verification of the Brown-Fisher-Hurtado theorem's hypotheses in the p-adic setting. We will revise the paper to include this check.","responses":[{"response":"We agree with the referee that an explicit check is required and that the original BFH proof uses features specific to the real Lie group setting. In the revised version, we will insert a dedicated paragraph or subsection immediately following the statement of the main theorem, where we verify that the relevant hypotheses hold or have suitable analogues for lattices in simple p-adic groups. Key points include: (1) the role of Cartan subgroups is played by maximal split tori, which exist and act with the necessary hyperbolicity properties in the p-adic case; (2) the derivative cocycle dynamics can be analyzed using the p-adic valuation and the action on the projective space over the p-adic field; (3) while connectedness is absent, the proof relies on the existence of elements with contracting/expanding behavior on the tangent bundle, which is guaranteed by the rank assumption and the structure of S-arithmetic lattices. We will also note that the extension to totally disconnected S-arithmetic groups follows by reducing to the simple factors via the maximal rank condition. This revision will make the applicability transparent.","revision_made":"yes","referee_comment":"[Abstract / main argument] The manuscript invokes the Brown-Fisher-Hurtado theorem without an explicit check that its hypotheses are satisfied for lattices in simple p-adic groups. In particular, the original BFH argument relies on the existence of Cartan subgroups, real-rank dynamics on the derivative cocycle, and connectedness properties that are absent or require replacement in the totally disconnected topology of groups such as SL(n, Q_p). Because the entire finiteness statement rests on this applicability, the omission is load-bearing."}],"tokens_in":1144,"tokens_out":392,"duration_ms":23343,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper proves that C^1 actions of lattices in simple p-adic groups on compact manifolds are finite when the manifold dimension is less than the rank, and extends the statement to lattices in totally disconnected S-arithmetic groups with the critical dimension being the max rank of the factors. This is the actual new content.\n\nIt takes the Brown-Fisher-Hurtado finiteness theorem and carries the conclusion over to these new classes of groups. The clean statement that keeps the same dimension threshold is useful for anyone tracking rigidity results in this area.\n\nThe soft spot is the reliance on BFH machinery. That work was built for connected real semisimple Lie groups and uses features like Cartan subgroups and connected-component dynamics on the derivative cocycle. P-adic groups are totally disconnected, so the paper has to show either that the original arguments still apply or that suitable replacements exist. The abstract says it uses the machinery, but without seeing the details of how the adaptation is handled, it is hard to judge whether the proof goes through.\n\nThis is for specialists working on group actions on manifolds and Zimmer-type rigidity. A reader already following BFH results will get a clear new statement to consider.\n\nI would send it to peer review. Referees who know the original BFH paper can check whether the extension is properly justified or whether extra arguments are required for the p-adic case.","headline":"Extends BFH finiteness to p-adic and S-arithmetic lattices under the same dimension bound, but the adaptation of connected-group tools needs explicit verification.","tokens_in":2135,"tokens_out":360,"would_cite":false,"duration_ms":18209,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Lattices in simple p-adic groups have only finite C1 actions on compact manifolds of dimension less than the rank.","keywords":["lattice actions","p-adic groups","S-arithmetic groups","C1 diffeomorphisms","compact manifolds","finiteness","group actions"],"falsifier":"A single non-finite C1 action of a lattice in a simple p-adic group on a compact manifold whose dimension is strictly less than the rank would disprove the claim.","tokens_in":2447,"feed_emoji":"","tokens_out":668,"duration_ms":26773,"temperature":0.7,"pith_summary":"The paper shows that actions by C1 diffeomorphisms of a lattice in a simple p-adic group on a compact manifold are finite whenever the manifold dimension is smaller than the rank of the group. It generalizes this to lattices in totally disconnected S-arithmetic groups, taking the critical dimension to be the largest rank among the simple factors involved. Sympathetic readers would be interested because the result limits the ways these groups can act smoothly and continuously on manifolds, forcing any such action to have finite image in the diffeomorphism group. The argument adapts techniques previously used for other groups to these p-adic and S-arithmetic settings.","feed_headline":"Lattices in p-adic groups act finitely below rank dimension","feed_subtitle":"Any C1 action on a compact manifold must be finite if dimension is less than the group's rank, extending to S-arithmetic cases.","key_machinery":"The Brown-Fisher-Hurtado machinery for proving finiteness of group actions on manifolds, applied here to lattices in p-adic and S-arithmetic groups.","core_discovery":"We prove that an action by C^1 diffeomorphisms of a lattice in a simple p-adic group on a compact manifold is finite, provided the dimension is less than the rank. We extend this statement to lattices in totally disconnected S-arithmetic groups, where the critical dimension is the maximal rank of the simple factors. This uses the machinery developed by Brown, Fisher, and Hurtado.","pith_inferences":["The same approach might yield finiteness statements for lattices in other totally disconnected groups if the machinery can be verified for them.","It remains open whether non-finite actions appear precisely when dimension equals the critical rank.","The theorem constrains possible smooth realizations of these groups as subgroups of diffeomorphism groups in low dimensions."],"forward_implications":["Any such action on a manifold of dimension below the rank must have finite image.","The same finiteness conclusion holds for lattices in totally disconnected S-arithmetic groups when dimension is below the maximal rank of the simple factors.","The result gives no information on actions whose dimension meets or exceeds the critical rank value.","Finiteness follows directly once the dimension hypothesis and the applicability of the cited machinery are granted."],"fun_headline_variants":["P-adic lattices restrict to finite C1 manifold actions below rank","Finite actions by S-arithmetic group lattices under maximal factor rank","C1 diffeos of p-adic lattices on manifolds finite below dimension rank","Lattice actions on compact manifolds finite for S-arithmetic groups below rank"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The Brown-Fisher-Hurtado machinery applies to lattices in simple p-adic groups and totally disconnected S-arithmetic groups to establish finiteness under the dimension condition.","fun_headline_variants_meta":{"raw":{"variants":["P-adic lattices restrict to finite C1 manifold actions below rank","Finite actions by S-arithmetic group lattices under maximal factor rank","C1 diffeos of p-adic lattices on manifolds finite below dimension rank","Lattice actions on compact manifolds finite for S-arithmetic groups below rank"]},"model":"grok-4.3","cost_usd":0.004072,"raw_usage":{"total_tokens":1994,"prompt_tokens":515,"num_sources_used":0,"completion_tokens":74,"cost_in_usd_ticks":40724500,"prompt_tokens_details":{"text_tokens":515,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1405,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":515,"tokens_out":74,"duration_ms":11337,"temperature":1.0,"reasoning_tokens":1405,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-02T05:24:05.421063+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A single non-finite C1 action of a lattice in a simple p-adic group on a compact manifold whose dimension is strictly less than the rank would disprove the claim.","supporting_citations":[],"review_version":1}