EXCERCISES, LIST 1 k-large and k-systolic complexes

1. Full subcomplex of a k-large complex is k-large.

2. Link of any simplex in a k-large complex is k-large.

3. (k-largeness in terms of local geodesics) A closed local geodesic of length n in a simplicial complex X is a sequence v_{0}, v_{1}, …, v_{n} of vertices in X such that v_{0} = v_{n}, d_{X}(v_{i}, v_{i+1}) = 1 and d_{X}(v_{i}, v_{i+2}) = 2 for all i (where indices are viewed mod n). Show that X is k-large iff there is no closed local geodesic of length < k in X.

4. By a systole of a simplicial complex X, denoted sys(X), we mean the length of the shortest cycle in X which is a full subcomplex of X. Show that, for k ≥ 4, X is k-large iff sys(X) ≥ k and ∀_{σ⊂X} sys(X_{σ}) ≥ k (we do not assume flagness here!).

5. (Missing part of proof of semi-local characterization of k-largeness, k ≥ 6) If for some k ≥ 6 X is locally k-large, and if sys_{h}(X) ≥ 4 then X is flag.

6. For k ≥ 6, X is k-large iff sys_{h}(X) ≥ k and ∀_{σ⊂X} sys_{h}(X_{σ}) ≥ k. Hint: induction.

7. For k ≥ 6, if X is simply connected and all vertex links X_{v} are k-large, then X is k-systolic. Hint: express links at simplices in terms of vertex links.

8. Barycentric subdivision of a 6-large (systolic) complex is in general not 6-large (not systolic, respectively).

9. The (boundary triangulation of) icosahedron is 5-large.

10. (Piotr Przytycki) The (boundary triangulation of) 600-cell is 5-large.

11. There is X which is flag, locally 5-large, with sys_{h}(X) ≥ 5 and such that X is not 5-large.

12. Product of a tree and a line has a systolic triangulation.

13. Let F be an orientable surface, F ≠ S^{2}. Then there is a 6-large triangulation of F.

14. Show that the following groups are systolic: Z ⊕ Z, other surface groups, Z ⊕ F_{2}.

15. (Tomek Elsner and Piotr Przytycki) Show that product of two finitely generated free groups is a systolic group. Hint: construct a 3-dimensional systolic complex for this group.

16. (Frederic Haglund) Recall that a vertex link of a cubical complex is a simplicial complex. Show that any simply connected cubical complex X of dimension not greater than 3, with 6-large vertex links, can be subdivided without introducing new vertices, into a systolic simplicial complex.

17. Fill the details of the proof that if X is 7-systolic then X^{(1)} is δ-hyperbolic. In particular, show that each bigon is 1-thin.

18. Every simplicial map f from triangulated 2-sphere S^{2} to systolic complex X (f : S^{2} → X) can be extended to a simplicial map F : B^{3} → X, where B^{3} is a triangulated 3ball with no internal vertices such that ∂B^{3} = S^{2}. Hint: use ”filling short cycles” argument.

19. Prove the following characterization of the class of finite ∞-large simplicial complexes: It is precisely the smallest class C such that:

(i) single simplex of arbitrary dimension belongs to C (ii) if X is obtained from some complexes X_{1}, X_{2} ∈ C by gluing them along a single simplex, then X ∈ C

20. Elaborate the following example: For any integer m ≥ 3 consider the Coxeter group

W_{m} = ⟨s_{1}, s_{2}, s_{3}|s_{1}^{2}, s_{2}^{2}, s_{3}^{2}, (s_{1}s_{2})^{m}, (s_{1}s_{3})^{m}, (s_{2}s_{3})^{m}⟩

and its Coxeter complex Σ_{m} being the triangulation of the plane with 2m triangles around every vertex. By residual finiteness of W_{m}, there is a torsion free normal subgroup N < W_{m} of finite index, such that the quotient surface N\Σ_{m} is 2m-large.

Comment: It is possible show that there exists a simply connected 3-dimensional pseudo-manifold X = X(m, N) with all vertex links isomorphic to N\Σ_{m}. It is obtained as the universal development of the 3-simplex of groups with groups ℤ_{2} at 2-faces, with dihedral groups D_{m} (of order 2m) at edges, and with the quotient groups W_{m}/N at vertices. For big enough m this pseudo-manifold is CAT(0) for the standard piecewise euclidean metric.

21. (Flat Geodesic Triangle Lemma by Tomek Elsner and Piotr Przytycki) Let a, b, c be arbitrary vertices in a systolic complex X. Show that there exit three geodesics in the 1-skeleton of X, [a, b], [b, c] and [a, c] which together bound a flat regular triangle in X. More precisely, these geodesics have the following property. There are vertices p, q, r with geodesics [p, q], [q, r], [r, p] whose union forms a cycle in X that bounds a triangulated regular flat triangle, and such that p ∈ [a, b] ∩ [b, c], q ∈ [b, c] ∩ [a, c] and r ∈ [a, c] ∩ [b, c].

22. (Tomek Elsner) Let X be systolic, and let Y ⊂ X be a full subcomplex such that the 1-skeleton Y^{(1)} isometrically embeds in X^{(1)}. Then Y is systolic.

A face complex of a cell polytopal complex Y is a simplicial complex Φ(Y) whose vertex set is the set of all cells (faces) in Y and in which a set of vertices spans a simplex iff the cells in Y corresponding to those vertices are all contained in a single cell of Y.

23. (Frederic Haglund) Show that if X is a k-large simplicial complex then its face complex Φ(X) is also k-large. Moreover, if X is k-systolic then so is Φ(X). Show also the converse implications.

24. Show that if Y is a simply connected cubical (or more generally simple polytopal) complex with k-large links then the face complex Φ(Y) is k-systolic.