EXCERCISES, LIST 2 3-convexity, convexity

1. Res(σ, X) is full in any flag X. Res(σ, X) is 3-convex in any 6-large X.

For a subcomplex A ⊂ X define B_{1}(A, x) := ∪{τ ⊂ X|τ ∩ A ≠ ∅}.

2. Show (directly from the definition) that for any simplex σ in a 6-large complex X the ball B_{1}(σ, X) is 3-convex in X.

3. Let Q_{1} ⊂ X_{1}, Q_{2} ⊂ X_{2} be 3-convex (convex) subcomplexes in 6-large (systolic) complexes. Suppose Q_{1}, Q_{2} are isomorphic and let f : Q_{1} → Q_{2} be a simplicial isomorphism. Then X_{1} ∪_{f} X_{2} is 6-large (systolic).

4. Let C be a clique in a 6-large simplicial complex X (i.e. a subcomplex spanned by infinite set of vertices in which every finite subset spans a simplex of X). Prove that:

(1) C is 3-convex in X

(2) the residue Res(C, X) is 3-convex in X

(3) the neighbourhood B_{1}(C, X) is 3-convex in X.

5. Any convex subcomplex of a systolic complex is (a) full

(b) 3-convex.

A subcomplex Y in a simplicial complex X is geodesically convex if any geodesic (in the 1-skeleton of X) connecting vertices of Y is fully contained in Y.

6. (Frederic Haglund) Let Y be a subcomplex in a systolic complex X. Prove that if Y is convex then it is geodesically convex. Prove also the converse implication, under assumption that Y is flag. Hint: to get second part use diagrammatics.

7. Prove that if γ is a geodesic in a 7-systolic complex X then the 1-neighbourhood of γ is convex. (Try first for 8-systolic X). Show that this is in general not true in 6-systolic X.

8. Intersection of any family of convex subcomplexes is (a) connected

(b) convex.

9. Show that the notion of convex hull is well defined in systolic complexes. Show that convex hull of a bounded subcomplex is bounded.

10. Let G be a finite group of automorphisms of a systolic complex X. Show that there is a bounded convex subcomplex of X which is G-invariant.

11. (Tomek Elsner) Check that a flat in a systolic complex (i.e. an equilaterally triangulated plane with isometrically embedded 1-skeleton) need not be a convex subcomplex (it need not be even quasi-convex).

12. (2-dimensional Helly) Let X be a systolic complex. Show that, given 4 convex complexes any three of which intersect, there is a simplex of X which intersects all four complexes.