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arxiv: 2604.06968 · v1 · submitted 2026-04-08 · 🧮 math.GR

A criterion for Tits alternative on the centralizer of a matrix

Adem Zeghib This is my paper

Pith reviewed 2026-05-10 17:12 UTC · model grok-4.3

classification 🧮 math.GR
keywords centralizerGL(n,Z)Tits alternativepolycyclic groupsconjugacy problemarithmetic groupssymplectic groups
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The pith

A matrix in GL(n, Z) has a polycyclic centralizer if and only if it satisfies a specific algebraic condition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper supplies a necessary and sufficient condition on a matrix so that its centralizer inside GL(n, Z) is polycyclic. This condition is equivalent to the centralizer containing no non-abelian free subgroup and therefore supplies an effective Tits alternative for these particular centralizers. A simpler condition on the matrix is also stated that forces the centralizer to be abelian. The same criteria are then used to obtain an effective solution to the conjugacy problem inside arithmetic groups that preserve a non-degenerate rational bilinear form, including the integral symplectic groups, by reducing the problem to known algorithms in GL(n, Z) and in polycyclic groups.

Core claim

We give a necessary and sufficient condition on a matrix for its centralizer in GL(n,Z) to be polycyclic, or equivalently in this case, not to contain a non-abelian free subgroup. We give a simple condition on the matrix ensuring that it is abelian. This can be thought of as an effective Tits alternative on centralizers in GL(n,Z). We apply these criteria to the conjugacy problem in certain arithmetic groups preserving a non-degenerate Q-bilinear form, such as integral symplectic groups. We derive an effective solution to the conjugacy problem in such groups when given matrices satisfy the above criterion.

What carries the argument

The algebraic criterion on the matrix that forces its centralizer in GL(n, Z) to be polycyclic rather than to contain a non-abelian free subgroup.

If this is right

  • The conjugacy problem admits an effective solution in the indicated arithmetic groups whenever the input matrices satisfy the criterion.
  • Existing algorithms for the conjugacy problem in GL(n, Z) and for orbit problems in polycyclic groups become directly applicable to these centralizers.
  • Under the simpler stated condition the centralizer is abelian.
  • The Tits alternative becomes effective for centralizers of matrices in GL(n, Z).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same style of criterion might extend to centralizers inside other arithmetic subgroups of GL(n, R) or over different base rings.
  • The reduction technique could be reused for membership or generation problems inside these centralizers.
  • The result shows that the linear-algebraic data of the matrix fully controls the virtual solvability of its integer centralizer.

Load-bearing premise

The polycyclicity of these centralizers is equivalent to the absence of non-abelian free subgroups and the existing algorithms for conjugacy and orbit problems apply directly once the criterion holds.

What would settle it

A concrete matrix in GL(n, Z) whose centralizer is polycyclic yet contains a non-abelian free subgroup, or a matrix that meets the stated criterion yet has a centralizer containing such a subgroup.

read the original abstract

We give a necessary and sufficient condition on a matrix for its centralizer in $\sf{GL}(n,\mathbb{Z})$ to be polycyclic, or equivalently in this case, not to contain a non-abelian free subgroup. We give a simple condition on the matrix ensuring that it is abelian. This can be thought of as an effective Tits alternative on centralizers in $\sf{GL}(n,\mathbb{Z})$. We apply these criteria to the conjugacy problem in certain arithmetic groups preserving a non-degenerate $\mathbb{Q}$-bilinear form, such as integral symplectic groups. We derive an effective solution to the conjugacy problem in such groups when given matrices satisfy the above criterion. This solution is based on effective solutions to the conjugacy problem in $\sf{GL}(n,\mathbb{Z})$ by Eick-Hofmann-O'Brien and to an orbit problem for polycyclic groups, by Eick and Ostheimer.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript states a necessary and sufficient condition on a matrix A ∈ GL(n,ℤ) such that its centralizer C_{GL(n,ℤ)}(A) is polycyclic (equivalently, contains no non-abelian free subgroup). It also supplies a simple sufficient condition on A guaranteeing that the centralizer is abelian. These criteria are then applied to obtain an effective solution to the conjugacy problem in arithmetic subgroups of GL(n,ℚ) that preserve a non-degenerate ℚ-bilinear form (e.g., integral symplectic groups), by reducing the problem to the known effective conjugacy algorithm in GL(n,ℤ) and the polycyclic orbit algorithm of Eick–Ostheimer.

Significance. If the stated criterion is correct, the work supplies a concrete, checkable condition under which the Tits alternative becomes effective for centralizers in GL(n,ℤ). The reduction to existing algorithms of Eick–Hofmann–O’Brien and Eick–Ostheimer makes the result immediately usable for computational conjugacy problems in groups such as Sp(2n,ℤ).

minor comments (3)
  1. The abstract asserts that the polycyclicity criterion is necessary and sufficient, yet the explicit matrix condition itself is not displayed; the main text should state the criterion (presumably in §3 or §4) in a form that can be checked directly from the rational canonical form or minimal polynomial of the matrix.
  2. The reduction to the Eick–Hofmann–O’Brien conjugacy algorithm and the Eick–Ostheimer orbit algorithm is invoked without an explicit verification that the centralizer, once polycyclic, satisfies the input requirements of those procedures (e.g., finite presentation or computable membership test). A short paragraph confirming these hypotheses would strengthen the claim.
  3. The phrase “or equivalently in this case” (abstract) linking polycyclicity to the absence of non-abelian free subgroups should be accompanied by a one-sentence reminder of why the Tits alternative applies to the centralizer subgroup.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript and for recommending minor revision. The referee's description correctly captures the main results on the polycyclicity criterion for centralizers in GL(n,Z) and the applications to the conjugacy problem in arithmetic groups preserving bilinear forms. Since the report lists no specific major comments, we have no points requiring direct response or revision at this stage. We remain available to incorporate any minor suggestions or clarifications if the editor or referee provides them.

Circularity Check

0 steps flagged

No circularity: criterion derived independently and applied via external algorithms

full rationale

The paper states a necessary and sufficient matrix criterion for the centralizer in GL(n,Z) to be polycyclic (hence free of non-abelian free subgroups via the Tits alternative invoked only 'in this case'). This criterion is presented as the main contribution and is not shown to be obtained by fitting parameters, self-definition, or reduction to prior self-citations. The subsequent application to the conjugacy problem explicitly reduces to independent external results (Eick-Hofmann-O'Brien conjugacy algorithm in GL(n,Z) and Eick-Ostheimer polycyclic orbit algorithm), which are cited as black-box effective procedures rather than derived within the paper. No load-bearing self-citation, ansatz smuggling, or renaming of known results occurs in the derivation chain; the argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard facts from linear algebra and the theory of polycyclic groups together with the Tits alternative; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption Centralizers in GL(n,Z) satisfy the Tits alternative and polycyclicity is equivalent to absence of non-abelian free subgroups for these groups.
    Stated explicitly in the abstract as the setting for the criterion.
  • standard math Existing algorithms for conjugacy in GL(n,Z) and orbit problems in polycyclic groups apply once the centralizer is known to be polycyclic.
    The application section invokes these prior results without re-deriving them.

pith-pipeline@v0.9.0 · 5454 in / 1421 out tokens · 53418 ms · 2026-05-10T17:12:36.976246+00:00 · methodology

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Reference graph

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