Notes of Eric Swenson’s Cambridge lecture 2-2-2017

Infinite torsion subgroups of ${CAT(0)}$ groups

Question. Let ${G}$ act geometrically on a proper ${CAT(0)}$ space. Can ${G}$ have an infinite torsion subgroup ?

Known. For cube complexes, Wise and Sageev show that a torsion group fixes a point (eventually at infinity). Tits alternative. So answer is no for these.

Coulon and Guirardel have an example of an infinite torsion group acting properly on an infinite dimensional ${CAT(0)}$ cube complex.

Today, as a kind of therapy, I collect all I know on this irritating open question.

1. ${CAT(0)}$ geometry

In ${CAT(0)}$ spaces, angles are well defined as limits of Euclidean comparison angles at small scales. Angle is always at most equal to the Euclidean comparison angle.

The visual boundary consists of equivalence classes of unit speed geodesic rays. Shadows of small balls seen from far away define a topology. If space ${X}$ is proper (need not be finite dimensional), ${X\cup\partial X}$ is compact and finite dimensional.

Angles in large biangles converge and define the angle metric on ${\partial X}$, that defines a finer topology. Let ${d_T}$ denote the corresponding path metric, known as the Tits metric. Angle and Tits coincide below level ${\pi}$. We denote by ${TX}$ the visual boundary equipped with Tits metric’s topology.

2. Conical limit points

If ${H, let ${\Lambda H}$, the limit set, denote the set of limit points of an orbit. A ray ${\alpha\in\Lambda H}$ is conical if some tubular neighborhood of ${\alpha}$ contains an unbounded subset of some orbit.

Theorem 1 If ${H has a conical limit point, then there is no bound on the orders of elements of ${H}$.

Indeed, there are numbers ${p_n\rightarrow\infty}$ and elements ${h_n\in H}$ such that ${h_n\alpha(p_n)\in \bar B(x_0,N)}$. Define rays

$\displaystyle \begin{array}{rcl} \alpha_n(t)=h_n(\alpha(t+p_n)),\quad t\in [-p_n,+\infty). \end{array}$

Up to extracting a subsequence, one can assume that ${\alpha_n}$ converge to a geodesic line ${L}$ uniformly on compact subsets. For ${j\gg i\gg 0}$, let ${g=h_i^{-1}h_j}$, and consider points ${\alpha(p_i)}$ (very close to ${g(\alpha(p_j))}$), ${g(\alpha(p_i))}$ (very close to ${g^2(\alpha(p_j))}$), and ${g^2(\alpha(p_i))}$ (very close to ${g^3(\alpha(p_j))}$). Hence the angle ${<_{g(\alpha(p_i))}(\alpha(p_i),g^2(\alpha(p_i)))}$ is almost ${\pi}$. Say ${g}$ has order ${n}$. Then iterates ${g^k(\alpha(p_i))}$ form the vertices of an ${n}$-gon with equal angles. In a ${CAT(0)}$ space, the sum of the angles of an ${n}$-gon is at most ${(n-2)\pi}$, this forces ${n}$ to be large.

Corollary 2 Every ${CAT(0)}$ group has a hyperbolic element.

Corollary 3 If ${H a ${CAT(0)}$ group is an infinite torsion group, then ${H}$ has no conical limit points.

3. Further results

Theorem 4 Let ${H acting geometrically on ${CAT(0)}$ space ${X}$. If

$\displaystyle \begin{array}{rcl} diam_T \Lambda H > 2\pi, \end{array}$

then ${H}$ contains a hyperbolic isometry.

Theorem 5 Let ${H acting geometrically on ${CAT(0)}$ space ${X}$. If ${H}$ is infinite torsion group, and if ${dim(X)}$ is minimal among such spaces, then ${\Lambda H}$ is infinite. Furthermore,

$\displaystyle \begin{array}{rcl} radius_T \Lambda H > \frac{\pi}{2}, \end{array}$

and ${H}$ does not fix a point in ${\partial X}$.