Groups and their finite shadows

Partly answered Do Kaplansky’s conjectures hold for torsion-free hyperbolic groups?

No in positive characteristic; in characteristic zero, units and zero divisors are open.

1. Kaplansky’s conjectures fail for torsion-free hyperbolic groups in positive characteristic

   Nidhal Mghirbi and Seth Douglas · version 1.0.3 · 1 November 2026 · 52 pages

   For every prime $p$, one finitely presented, torsion-free, word-hyperbolic group whose group algebra over every field of characteristic $p$ is not directly finite and has a non-trivial idempotent, a zero divisor and a non-trivial unit.

   • Extended Do Kaplansky’s group-ring conjectures hold for every torsion-free group?
   • Partly answered Do Kaplansky’s group-ring conjectures hold for torsion-free hyperbolic groups?

   The paperPDFDOICode

2. Finitely presented infinite $p$-groups for every prime

   Seth Douglas and Nidhal Mghirbi · version 1.0.1 · 30 October 2026 · 38 pages

   For every prime $p$, an infinite, residually finite group with an ordinary finite presentation in which every element has finite $p$-power order. One presentation, written over the integers, serves every prime.

   • Extended Can an infinite periodic group have an ordinary finite presentation?
   • Answered Is there a finitely presented infinite $p$-group for every prime?

   The paperPDFDOICode

3. Non-soficity of OpenAI’s torsion-free hyperbolic group

   Seth Douglas and Nidhal Mghirbi · version 1.1.0 · 14 October 2026

   OpenAI’s torsion-free word-hyperbolic group, built not to be residually finite, is not sofic: its rank argument survives approximation. Version 1.1.0 adds that its group rings over division rings of the construction’s characteristic have no Linnell division ring.

   • Answered Is OpenAI’s torsion-free hyperbolic group sofic?

   The paperPDFDOICode

4. Amenable groups with nearly exponential sofic profile, and quantum channels that need nearly linear memory

   Seth Douglas and Nidhal Mghirbi · version 1.2.0 · 2 November 2026 · 78 pages · arXiv:2610.04271

   A finitely presented elementary amenable group with a finite piece whose sofic profile is $\exp (r^{1+o(1)})$.

   • Answered Can a sofic group have a sofic profile that is unbounded and not linear?
   • Answered Can a finitely presented amenable group have nearly exponential sofic profile?

   The paperPDFDOICode

5. Houghton’s group $H_3$ has superpolynomial sofic profile

   Seth Douglas and Nidhal Mghirbi · version 1.2.0 · 2 November 2026 · 37 pages

   Houghton’s finitely presented elementary amenable group $H_3$ has a finite chunk whose sofic profile is superpolynomial.

   • Answered Can a sofic group have a sofic profile that is unbounded and not linear?
   • Answered Is the sofic profile of Houghton’s group $H_3$ polynomial?

   The paperPDFDOICode

• Extended Do Kaplansky’s group-ring conjectures hold for every torsion-free group?
• Extended Can an infinite periodic group have an ordinary finite presentation?
• Answered Can a sofic group have a sofic profile that is unbounded and not linear?