Infinitely presented simple groups separated by homological finiteness properties

2025/10/02 by Isenrich, Claudio Llosa, Schesler, Eduard, Wu, Xiaolei
#20E08 #20E32 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2510.01952

Abstract

Given a finitely generated linear group G over ℚ, we construct a simple group Γ that has the same finiteness properties as G and admits G as a quasi-retract. As an application, we construct a simple group of type FP_∞ that is not finitely presented. Moreover we show that for every n ∈ ℕ there is a simple group of type FP_n that is neither finitely presented nor of type FP_n+1. Since our simple groups arise as Röver--Nekrashevych groups, this answers a question of Zaremsky.

Citations

• Simple totally disconnected locally compact groups separated by finiteness properties
• On the Boone--Higman Conjecture for groups acting on locally finite trees
• Embedding groups into boundedly acyclic groups
• Boone--Higman Embeddings for Contracting Self-Similar Groups
• Embedding finitely presented self-similar groups into finitely presented simple groups
• Progress around the Boone-Higman Conjecture
• Finitely presented simple groups with at least exponential Dehn function
• Amenability and profinite completions of finitely generated groups
• Hyperbolic groups with almost finitely presented subgroups
• Simple groups separated by finiteness properties
• Filling functions of arithmetic groups
• Subgroups of almost finitely presented groups
• Compact presentability of tree almost automorphism groups
• Finitely presented groups associated with expanding maps
• The conjugacy problem in automaton groups is not solvable
• The conjugacy problem in automaton groups is not solvable
• Arithmetic aspects of self-similar groups

Cited by

• Simple totally disconnected locally compact groups separated by finiteness properties

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• Finite Germ Extensions
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• Presentations of finite simple groups: a quantitative approach