Systolic Groups (Math) - AI

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In mathematics, systolic groups are groups that act geometrically (properly discontinuously and cocompactly) on a systolic complex. These groups are a primary focus in geometric group theory because they exhibit a form of "simplicial non-positive curvature," making them combinatorial analogs of CAT(0) or non-positively curved spaces. [1, 2, 3, 4, 5]

Key Components
  • Systolic Complex: A connected, simply connected simplicial complex where the "links" of all its simplices are 6-large.
    • 6-large means the complex is "flag" (any set of vertices that are pairwise connected by edges spans a simplex) and has no embedded cycles of length less than 6 that are "full subcomplexes".
    • Essentially, every simplicial loop of length 4 or 5 must have a diagonal.
  • Geometric Action: For a group to be called systolic, it must act by simplicial automorphisms on one of these complexes such that the action is:
    • Proper: Each compact subcomplex is moved away from itself by all but a finite number of group elements.
    • Cocompact: There is a compact part of the complex that, when moved by all group elements, covers the entire complex. [1, 2, 3, 4, 5, 6]
Importance and Properties
Systolic groups have unique properties that set them apart from other non-positively curved groups: [1, 2, 3]
  • Biautomaticity: Every systolic group is biautomatic, which implies that the word and conjugacy problems for these groups are solvable.
  • Curvature Analogy: While they share properties with CAT(0) spaces, systolicity is a distinct combinatorial notion. For example, 7-systolic groups are always Gromov-hyperbolic.
  • Fixed Points: For any simplicial action of a finite group on a systolic complex, there is always an invariant simplex (a "little fixed point theorem").
  • Exotic Examples: This theory has been used to construct groups with unexpected properties in arbitrarily high dimensions. [1, 2, 3, 4, 5, 6, 7]

Would you like to explore a specific property of these groups, such as their biautomaticity or their relation to hyperbolic groups?

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