Extending the bump function of irregular geodesic rays to the Karpelevich compactification

In this very short post, we look at the properties of a certain bump function of the collection of all geodesic rays emanating from a fixed point $x_0\in X$ towards irregular points on the geodesic boundary $\partial X$ of a non-compact symmetric space

Assume we are given a non-compact symmetric spaces $X$ , and a point $x_0\in X$ . If $\gamma$ is a geodesic in $X$ with $\gamma(0) = x_0$ , let $U$ be a normal neighborhood of $\gamma(\R)\subset X$ constructed as the parallel translates along $\gamma$ of a fixed ball $B_r(x_0)$ . We can easily construct a bump function $s: X\to [0,1]$ with the following properties

the function $s$ only depends on the distance from $\gamma(\R)$ ,
the function $s$ is zero on $X\backslash U$ and $1$ on a neighborhood of $\gamma(\R)$ .

In the case $X = \H^2$ is the real hyperbolic plane, with the Poincare disc model and $x_0 = 0$ , then the geodesic ray from $0$ is just a straight line through the origin. The neighborhood $U$ is a tube around the geodesic line which shrinks in diameter as it gets closer to the boundary, where it collapses to a single point. If we assume the function $s$ could be extended to the closure of the disc (which is the geodesic compactification of $\H^2$ ), then we are faced with the following difficult question: What are the limit points of the set $s^{-1}(1/2)$ ?

Clearly, this set is unbounded in the hyperbolic metric, so by compactness it would have to have some limit point on the boundary of the disk. However, these points lie in $U$ and since $U$ collapses to a single point on the boundary, we conclude that $\overline{s^{-1}(1/2)}$ intersects the boundary at precisely two points, namely the limit points of $\gamma(\R)$ . Thus we find that by choosing what sequence in $\H^2$ converges to the boundary point of $\gamma(\R)$ , the “extension” of $s$ can attain any value between $0$ and $1$ on these boundary points. Thus the extension cannot be continuous.

In a slightly more general terms, let us try to create a bump like function for the collection of irregular boundary points. Recall that a point $v\in X(\infty)$ on the geodesic boundary is called irregular if the corresponding geodesic $[\gamma] = v$ is contain in at least two maximal totally geodesic submanifolds of $X$ .

Let ${X}$ be a non-compact symmetric space, let ${x_0\subset X}$ be any point. Let ${V\subset X}$ be the union of all geodesic rays from ${x_0}$ to ${\partial X_{irr}}$ (the irregular points on the geodesic boundary). Let ${U_r}$ denote a normal neighborhood of ${V}$ of radius ${r}$ in the metric of ${X}$ . This is just the union of all parallel translates of a fixed ball ${B_r(x_0)}$ along the irregular geodesic rays from ${x_0}$ . Let ${s: X\to [0,1]}$ be a continuous bump function which is $1$ on ${\overline{U}_{1/2}}$ and zero outside of ${U_1}$ and assume $s$ only depends on the distance from ${V}$ . The function $s$ is then a bump function for the set $V$ of geodesic rays from $x_0$ going towards irregular boundary points.

Let ${\phi: X\to \C}$ be given by

$\displaystyle \phi(x) = 1 - s(x)$

The function ${\phi}$ is clearly continuous and bounded on ${X}$ . Since it is a very natural construction, it may be interesting to look at what function we get on the boundary when we extend ${\phi}$ to certain compactifications of ${X}$ .

First, let us look at the the situation for the geodesic compactification. We will show the function ${\phi}$ converges radially to a well defined function on ${X\cup \partial X}$ which is ${1}$ on the regular boundary points and ${0}$ on the irregular boundary points. Though we should emphasize that the extension is radial, and not continuous. Continuous extensions cannot exist, as the next lemma shows

Lemma 1 Let ${X}$ be any manifold with boundary ${\partial X}$ and ${f \in C_b(X)}$ . Assume ${f}$ extends to a function ${\hat{f}}$ on ${X\cup \partial X}$ in the sense that for any sequence ${x_i \in X}$ , with ${x_i \to x_\infty \in \partial X}$ , we have ${\hat{f}(x_i) \to \hat{f}(x_\infty)}$ , then ${\hat{f}|_{\partial X}}$ is continuous.

Proof

Assume for simplicity ${X}$ is imbedded in some euclidean space ${\R^n}$ (using Whitney’s theorem) and give it the inherited distance function which is compatible with the topology of ${X}$ . Let ${v \in \partial X}$ be a point. Then ${\hat{f}|_{\partial X}}$ is continuous at ${v}$ if and only if for any sequence ${v_i \in \partial X}$ converging to ${v}$ , ${\hat{f}(v_i)}$ converges to ${\hat{f}(v)}$ . Let ${x_i \in X}$ be a sequence such that

${||x_n - v_n|| < 1/n}$
${|\hat{f}(x_n) - \hat{f}(v_n)| < 1/n}$

This can be done by continuity of ${\hat{f}}$ . Then it is easy to check that ${x_i}$ converges to ${v}$ . Since ${\hat{f}}$ is an extension of ${f}$ , we have ${\hat{f}(v) = \lim \hat{f}(x_i) = \lim \hat{f}(v_i) }$ . Since ${\lim f(x_i)}$ is assumed to be independent of choice of convergent sequence to ${v}$ , it follows that ${\lim \hat{f}(v_i)}$ exists and is independent of choice of sequence ${v_i}$ .

In summary, if ${f \in C(X)}$ is real valued and extends to a function on ${\overline{X}}$ which has a jump discontinuity at ${v}$ (from ${n}$ to ${m}$ in ${\R}$ ), then some sequence in ${f^{-1}((n + m)/2)}$ converges to ${v}$ . Hence the extension is not well defined at $v$ .

However, let us show our function $\phi$ converges to a well defined function on $X\cup \partial X_{reg}$ but not to the whole geodesic compactification $X\cup \partial X$ . To this end let

$\displaystyle U_r^n := \{ \xi \in \partial X ~|~ \xi = [\gamma], ~ \text{with } \gamma(0) = x_0, \text{ and } \gamma(n) \in U_r \}$

the shadow of ${U_r^n}$ in ${\partial X}$ . We have –

Lemma 2 The sets ${U_r^n \subset \partial X}$ are open neighborhoods of ${\partial X_{irr}}$ and

$\displaystyle \bigcap_{n\in \N} U_r^n = \partial X_{irr}.$

Proof (sketch)

There is a natural homeomorphism between ${\partial X}$ and the “sphere”

$\displaystyle S_t = \{x \in X ~|~ d(x, x_0) = n\}$

by sending a point ${y\in S_t}$ to the unique geodesic ray from ${x_0}$ to ${y}$ . The set ${U_n^t = U_n \cap S_t}$ is thus open in ${\partial X}$ .

For the last claim, we can argue as follows. If we parallel translate a ball ${B}$ of radius ${r}$ along a geodesic ray, the “size of ” ${B}$ as seen from a fixed observer goes to zero. Put differently, the maximal angle between geodesic rays from the observer to two points in the ball goes to zero as ${B}$ is translated away from the observer.

By the same reasoning the sets ${U_r\cap S_t}$ as seen from ${x_0}$ will get closer and closer to the set ${V\cap S_t}$ . Put differently, the maximal angle between geodesic rays from ${x_0}$ and ${\partial X_{irr} \subset T_{x_0}X}$ will tend to zero.

Now if ${x_i \in X}$ is a sequence converging to ${x_\infty \in \partial X_{reg}}$ , then, using the definition of the topology on ${X\cup \partial X}$ , if ${v_i \in T_{x_0}X}$ denotes the unit vector corresponding to the geodesic ray from ${x_0}$ to ${x_i}$ and ${v_\infty}$ the geodesic ray going to ${x_\infty}$ , then the ${v_i \to v_\infty}$ in ${T_{x_0}(X)}$ . Picking any open neighborhood ${O}$ of ${v_\infty}$ such that ${O\cap \partial X_{irr} = \emptyset }$ . Then for ${N}$ large enough we have ${v_i \in O}$ and ${U_r^i \cap O = \empty}$ for all ${i\geq N}$ , where the last part follow from the above lemma. It follows that the geodesic ray converging to ${x_\infty}$ is eventually outside of ${U_r}$ and ${\phi(x_i) \to 1}$ . So ${\phi}$ extends to ${\partial X_{reg}}$ . Note that the sequence ${x_i}$ was arbitrary in the above example, so ${\phi}$ extends continuously to ${X\cup \partial X_{reg}}$ .

However, we cannot extend continuously to ${\partial X_{irr}}$ . If ${x_i \to x_\infty \in \partial X_{irr}}$ , then ${v_i \in U_r^n}$ for some ${n}$ . But if ${|d(x_0, x_i)| > n}$ this does not imply ${x_i \in U_r}$ . It is actually possible to check that ${\phi}$ can converge to any value in ${[0, 1]}$ by converging along different sequences ${x_i}$ to the same irregular point. What we mean by extending ${\phi}$ to ${X\cup \partial X}$ is that we extend it along geodesic rays from ${x_0}$ . In this case, there is only one value ${\phi}$ can attain at any point in ${\partial X_{\infty}}$ and we are done.

1. Karpelevich compactification to the rescue

The function ${\phi}$ does however extend to a continuous function on the Karpelevich compactification ${\overline{X}^K}$ . Let’s see why. The book of Borel and Ji is our reference for the definitions of the Karpelevich compactification.

We will not define it here, but we will need to know that boundary points of ${\overline{X}^K}$ can be written as tuples ${(\xi, z)}$ , where ${\xi \in \partial X}$ and ${z\in \overline{X}_\xi}^K$ and $\overline{X}_\xi}^K$ is the Karpelevich compactification of the boundary symmetric space at ${\xi}$ , meaning if ${P_\xi = M_\xi A_\xi N_\xi}$ is the parabolic subgroup stabilizing ${\xi}$ , then

${X_\xi = M_\xi/(M_\xi\cap K)}.$

The space $X_\xi$ can be identified with the totally geodesic submanifold $M_\xi x_0 \subset X$ and its geodesic compactification is isomorphic to the closure of $M_\xi x_0$ in $X\cup \partial X$ . In case $X$ has rank $1$ the compactification $\overline{X}^K = X\cup X(\infty)$ by definition. Since $X_\xi$ has rank lower than $X$ , if $rk(X) = 2$ , the boundary $\partial \overline{X}^K$ can be expressed using only the geodesic compactification, which simplifies the exposition.

Assuming a sequence of points ${x_i \in X}$ converge to a boundary point ${\xi \in \partial X}$ . Then, one can always express ${x_i}$ as

$\displaystyle x_i = g_i a_i m_i$

where ${a_i \in A_\xi}$ , ${m_i}$ is a representative in ${X_\xi}$ and ${g_i \in G}$ are such that ${g_i \to e}$ (mentioned in the proof of Proposition I.15.6 of but also implicit in the definition of the Karpelevich compactification on p. 53).

The sequence $x_i$ converges to ${(\xi, z_\infty)}$ if and only if

${a_im_i \to \xi}$ in ${X\cup \partial X}$ or equivalently ${x_i \to \xi}$ in ${X\cup \partial X}$ .
${m_i \to z_\infty}$ in ${\overline{X}^K_\xi}$ .

Now ${x_i \in X}$ be a sequence converging to ${(v, z)}$ where ${v\in \partial X}$ and ${z\in \overline{X}_v^K = X_v\cup \partial X_v}$ (recall that we assume $X$ has rank 2!). Let ${\gamma_v}$ be the geodesic centered at ${x_0}$ in the direction of ${v}$ , i.e. $\gamma(0) = x_0$ and $\gamma_v'(0) = v$ . Recall that $A = A_v$ is the Cartan subgroup of $P_v$ , hence ${v}$ is a limit point of ${A}$ in ${\partial X}$ , and we can write ${\gamma_v(t) = exp(tH) x_0}$ for some ${H \in \mathfrak{a}}$ (if not we would have to write ${\gamma_v(t) = k \gamma(tH)}$ for some ${k\in K}$ using the polar decomposition of ${X}$ described in a previous blog post).

$\begin{align*} d(x_i, \gamma_v) &:= \inf_t d(g_i a_i m_i x_0, exp(tH)x_0) \\& \to \inf_t \inf_t d(a_i m_i x_0, exp(tH)x_0) \\&:= \inf_t \inf_t d(m_i x_0, x_0) \end{align*}$

since ${g_i \to e}$ (so ${d(g_i y, y) \to 0}$ uniformly in ${y}$ ) and ${Mx_0}$ is a totally geodesic submanifold of ${X}$ orthogonal to ${\gamma_v}$ at ${x_0}$ . Since ${m_i}$ converges in the geodesic compactification of ${X_v = M_v/(M_v\cap K) = M_v x_0}$ it follows that ${d(m_ix_0, x_0)}$ converges or diverges to ${\infty}$ . Either way the the distance ${d(x_i, \gamma_v)}$ converges in $\R\cup \{\infty\}$ and if ${y_i}$ is another sequence converging to ${v}$ with

$\displaystyle \lim_i d(y_i, \gamma_v) \neq \lim_i d(x_i, \gamma_v)$

then ${\lim y_i \neq \lim x_i}$ in ${\partial \overline{X}^K}$ as the limit keeps track of the distance from ${\gamma_v}$ . This is all we need to ensure that ${\phi(v, z) = \lim \phi(x_i)}$ is well defined and does not depend on the choice of convergent sequence in ${X}$ . We have thus proved

Lemma 3 The function $\phi$ extends to a continuous function on the Karpelevich compactification.

Heuristically, what is happening in the Karpelevich compactificaiton is that we have taken the points on $\partial X(\infty)$ where the function $s$ (or $\phi$ ) could not be extended and “blown” them up by gluing in the boundary symmetric spaces $\overline{X}^K_\xi$ at these points. The extension of the function $s$ is then a bump function of some open set in $X_\xi \subset \partial \overline{X}^K$ where $\xi$ runs over all limit points of irregular boundary points.

Szabo, G., Wu, J., & Zacharias, J. (2017). Rokhlin dimension for actions of residually finite groups. ArXiv:1408.6096 [Math]. http://arxiv.org/abs/1408.6096

Guentner, E., Willett, R., & Yu, G. (2015). Dynamic Asymptotic Dimension: relation to dynamics, topology, coarse geometry, and $C^*$-algebras. ArXiv:1510.07769 [Math]. http://arxiv.org/abs/1510.07769

Conley, C., Jackson, S., Marks, A., Seward, B., & Tucker-Drob, R. (2020). Borel asymptotic dimension and hyperfinite equivalence relations. ArXiv:2009.06721 [Math]. http://arxiv.org/abs/2009.06721

Bell, G., & Dranishnikov, A. (2007). Asymptotic Dimension. ArXiv:Math/0703766. http://arxiv.org/abs/math/0703766

doi:10.1016/j.topol.2008.02.011 | Elsevier Enhanced Reader. (n.d.). https://doi.org/10.1016/j.topol.2008.02.011

Serre, J.-P. (1980). Trees and Amalgams. In J.-P. Serre, Trees (pp. 1–68). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-61856-7_1

Hill, T. P. (2021). An Evolutionary Theory for the Variability Hypothesis. ArXiv:1703.04184 [q-Bio]. http://arxiv.org/abs/1703.04184

Albuquerque, P. (1999). Patterson-Sullivan Theory in Higher Rank Symmetric Spaces. Geometric & Functional Analysis GAFA, 9(1), 1–28. https://doi.org/10.1007/s000390050079

Kadison, R. V., & Ringrose, J. R. (1972). Cohomology of operator algebras. III : reduction to normal cohomology. Bulletin de la Société Mathématique de France, 100, 73–96. https://doi.org/10.24033/bsmf.1731

Kadison, R. V., & Ringrose, J. R. (1971). Cohomology of operator algebras II. Extended cobounding and the hyperfinite case. Arkiv För Matematik, 9(1–2), 55–63. https://doi.org/10.1007/BF02383637

Kadison, R. V., & Ringrose, J. R. (1971). Cohomology of operator algebras: I. Type I von Neumann algebras. Acta Mathematica, 126(none), 227–243. https://doi.org/10.1007/BF02392032

Pop, F., & Smith, R. R. (2010). Vanishing of second cohomology for tensor products of type II1 von Neumann algebras. Journal of Functional Analysis, 258(8), 2695–2707. https://doi.org/10.1016/j.jfa.2010.01.013

Christensen, E., Pop, F., Sinclair, A., & Smith, R. (2003). Hochschild Cohomology Of Factors With Property Gamma. Annals of Mathematics, 158, 635–659.

Hochschild cohomology of factors with property $\Gamma$ | Annals of Mathematics. (n.d.). Retrieved September 4, 2021, from https://annals.math.princeton.edu/2003/158-2/p07

Cameron, J. (2009). Hochschild cohomology of II1 factors with Cartan maximal abelian subalgebras. Proceedings of the Edinburgh Mathematical Society, 52, 287–295. https://doi.org/10.1017/S0013091507000053

Cameron, J. (2009). Hochschild cohomology of II1 factors with Cartan maximal abelian subalgebras. Proceedings of the Edinburgh Mathematical Society. https://doi.org/10.1017/S0013091507000053

Johnson, B. E., Kadison, R. V., & Ringrose, J. R. (1972). Cohomology of operator algebras. III : reduction to normal cohomology. Bulletin de La SociéTé MathéMatique de France, 79, 73–96. https://doi.org/10.24033/bsmf.1731

Sinclair, A. M., & Smith, R. R. (1998). Hochschild Cohomology for Von Neumann Algebras with Cartan Subalgebras. American Journal of Mathematics, 120(5), 1043–1057. https://www.jstor.org/stable/25098635

Qian, W., & Shen, J. (2014). Hochschild cohomology of type II$_1$ von Neumann algebras with Property $\Gamma$. ArXiv:1407.0664 [Math]. http://arxiv.org/abs/1407.0664

Sinclair, A. M., & Smith, R. R. (1995). Hochschild Cohomology of Von Neumann Algebras. Cambridge University Press.

Hochschild Cohomology of Von Neumann Algebras | Abstract analysis. (n.d.). Cambridge University Press. Retrieved September 4, 2021, from https://www.cambridge.org/it/academic/subjects/mathematics/abstract-analysis/hochschild-cohomology-von-neumann-algebras, https://www.cambridge.org/it/academic/subjects/mathematics/abstract-analysis

Eynard, B. (2014). A short overview of the “Topological recursion.” ArXiv:1412.3286 [Hep-Th, Physics:Math-Ph]. http://arxiv.org/abs/1412.3286

Pimsner, M. V. (1986). KK-groups of crossed products by groups acting on trees. Inventiones Mathematicae, 86(3), 603–634. https://doi.org/10.1007/BF01389271

Pimsner, M. (1986). KK-groups of crossed products by groups acting on trees. https://doi.org/10.1007/BF01389271

Rieffel, M. (1982). Applications of strong Morita equivalence for transofrmation C*-algebras. 38(1). https://math.berkeley.edu/~rieffel/papers/applications.pdf

Kloeckner, B. (2010). Symmetric spaces of higher rank do not admit differentiable compactifications. Mathematische Annalen, 347(4), 951–961. https://doi.org/10.1007/s00208-009-0464-z

Paradan, P.-E. (2009). Symmetric spaces of the non-compact type: Lie groups. Séminaires et Congrès, Géométries à courbure négative ou nulle, groupes discrets et rigidités(18), 39–76. https://hal.archives-ouvertes.fr/hal-00773255

Connes, A., & Skandalis, G. (1984). The Longitudinal Index Theorem for Foliations. Publications of the Research Institute for Mathematical Sciences, 20(6), 1139–1183. https://doi.org/10.2977/prims/1195180375

Schroeder, M. (2010). Patterson-Sullivan distributions for symmetric spaces of the noncompact type. ArXiv:1012.1113 [Math]. http://arxiv.org/abs/1012.1113

Helgason, S. (1979). Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press.

Emerson, H., & Meyer, R. (2006). Euler characteristics and Gysin sequences for group actions on boundaries. Mathematische Annalen, 334(4), 853–904. https://doi.org/10.1007/s00208-005-0747-y

Blackadar, B. (1986). K-Theory for Operator Algebras. Springer-Verlag. https://doi.org/10.1007/978-1-4613-9572-0

Ji, L., & Macpherson, R. (2002). Geometry of compactifications of locally symmetric spaces. Annales de l’institut Fourier, 52(2), 457–559. https://doi.org/10.5802/aif.1893

Ulsnaes, T. (n.d.). Note series.

Ulsnaes, T. (2021). The Karpelevic compactification.

(N.d.).

Jin, X. (2021). Homological Mirror Symmetry for the universal centralizers I: The adjoint group case. ArXiv:2107.13395 [Math-Ph]. http://arxiv.org/abs/2107.13395

Soule, C. (2004). An introduction to arithmetic groups. https://hal.archives-ouvertes.fr/hal-00001348

Rosenberg, J. M., & Schochet, C. (1986). The Künneth theorem and the universal coefficient theorem for equivariant K-theory and KK-theory. American Mathematical Society.

Rieffel, M. A. (2002). Group C∗-Algebras as Compact Quantum Metric Spaces. Documenta Mathematica, 48.

Ulsnaes, T. (n.d.). Compactifications of symmetric spaces and their applications. X ., 52.

Ulsnaes, T. (n.d.). An example of the index of a Rockland operator on a contact manifold. 68.

Ulsnaes, T. (n.d.). A simple proof of the Hard Lefschetz theorem. 37.

Ulsnaes, T. (n.d.). Pensum MATH4450 Videreg˚aende Funksjonal analyse V˚ar 2017. 22.

Videregående Funksjonalanalyse. (n.d.). Retrieved June 22, 2021, from https://www.overleaf.com/project/59920ace6080cd3eb133870e

1. Karpelevich compactification to the rescue

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