The Michigan Mathematical Journal

A topological characterization of the underlying spaces of complete R-trees

Paul Fabel

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Source
Michigan Math. J., Volume 64, Issue 4 (2015), 881-887.

Dates
First available in Project Euclid: 18 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1447878035

Mathematical Reviews number (MathSciNet)
MR3426619

Zentralblatt MATH identifier
1345.54016

Subjects
Primary: 54D05: Connected and locally connected spaces (general aspects) 54F50: Spaces of dimension $\leq 1$; curves, dendrites [See also 26A03]
Secondary: 54E35: Metric spaces, metrizability

Citation

Fabel, Paul. A topological characterization of the underlying spaces of complete R-trees. Michigan Math. J. 64 (2015), no. 4, 881--887. https://projecteuclid.org/euclid.mmj/1447878035


Export citation

References

  • \item[(1)] There exists a compatible metric $d$ such that $(P,d)$ is a complete R-tree.
  • \item[(2)] There exists a compatible metric $d$ such that $(P,d)$ is an R-tree and such that $(P,d)$ is an open subspace of its metric completion $\overline{ (P,d)}$.
  • \item[(3)] $P$ is metrizable, locally path connected, uniquely arcwise connected, and locally interval compact. \endenumerate \endtheorem \sectionPreliminaries, Examples, Remarks, and Lemmas An arc is a single point or a space homeomorphic to $[0,1]$. A $p$ -based topological R-tree $(P,\tau,p,\leq,\hat{\ }\,)$ is a metrizable, uniquely arcwise connected, locally path connected space with $p\in P$ and $ [x,y]\subset P$ denoting the unique arc from $x$ to $y$. The space $P$ enjoys both the associative binary operation $\hat{\ }$ such that $[p,x\,\hat{\ }\, y]=[p,x]\cap[ p,y]$ and the partial order $\leq$ such that $y\leq x$ iff $y\in[ p,x]$. Notationally, we may suppress $\leq$ and $\hat{\ }$ if it is understood that $p$ is the basepoint, and $\tau$ can be replaced by $d$ or $D$ if $P$ is equipped with the particular metric $d$ or $ D$. A metric space $(P,d)$ is complete if each Cauchy sequence has a limit, and we remind the reader that every metric space can be embedded as a dense subspace of a complete metric space [Munkres?], uniquely up to isometry. \beginexample \labelbadLet $P$ denote the planar subspace $([0,1]\times\{0\})\cup (\bigcup_{n=1}^{\infty}\{\frac{1}{n}\}\times[ 0,\frac{1}{n}))$. Note that $P$ is not the underlying space of a complete R-tree since the half open intervals $\{\frac{1}{n}\}\times[ 0,\frac{1}{n})$ would be forced to have infinite geometric length, violating the topological fact that $ x_{n}\rightarrow0$ if $x_{n}\in\{\frac{1}{n}\}\times[ 0,\frac{1}{n} )$. Note that $P$ is a $G_{\delta}$ subspace of the plane (the intersection of countably many open planar sets), and hence $P$ is topologically complete. \endexample The following fact follows easily from the algebraic properties of $(P, \hat{\ },\leq)$. \beginlemma \labelhatSuppose $(P,p,\tau\leq,\hat{\ }\,)$ is a p-based topological R-tree and $[p,z]\cap[ x,y]=\emptyset$. Then $x\,\hat{\ }\,z=y\,\hat{\ }\,z$. \endlemma \beginproof Note that $a\,\hat{\ }\,b\leq b$ since $a\,\hat{\ }\,b\in[ p,b]$ and $ a\leq b\Rightarrow a\,\hat{\ }\,b=a$ since $[p,a]\cap[ p,b]=[p,a]=[p,a\,\hat{\ }\,b]$. Note $\{x\,\hat{\ }\,z,x\,\hat{\ }\,y\}\subset[ p,x]$ and $x\,\hat{\ }\,z<x\,\hat{\ }\,y$ (since otherwise we obtain the contradiction $x\,\hat{\ }\,z\in[ p,z]\cap [ x\,\hat{\ }\,y,x]\subset[ p,z]\cap[ x,y]$). By a symmetric argument we conclude $y\,\hat{\ }\,z<y\,\hat{\ }\,x$. Thus, $ \{x\,\hat{\ }\,z,y\,\hat{\ }\,z\}\subset[ p,x\,\hat{\ }\,y]$. Note that $y\,\hat{\ }\,z\in[ p,x]\cap[ p,z]$ and thus $y\,\hat{\ }\,z\leq x\,\hat{\ }\,z$. By a symmetric argument, $x\,\hat{\ }\,z\leq y\,\hat{\ }\,z$, and thus $x\,\hat{\ }\,z=y\,\hat{\ }\,z$. \endproof The following lemma is also a consequence of the fact that the metric completion of an R-tree is an R-tree [Imrich,Chiswell?]. \beginlemma \labeldivergeSuppose $(P,d,p,\leq,\hat{\ }\,)$ is an incomplete $p$ -based R-tree with metric completion $\overline{(P,d,p,\leq,\hat{\ }\,)} $. Suppose $y\in\partial P=\overline{(P,d)}\setminus P$. There exists an order-preserving isometric embedding $h:[0,d(x,y))\rightarrow(P,d)$ such that $h(0)=p$ and $y=\lim_{t\rightarrow d(x,y)}h(t)$. In particular, since the compactum $h([0,d(x,y)])$ is closed in the metric space $\overline{(P,d)}$, $h([0,d(x,y))=P\cap h([0,d(x,y)])$ is a closed subspace of $P$. \endlemma \beginproof Obtain a sequence $z_{n}\in P$ with $d(z_{n},y)\rightarrow0$. For each $ N\in\{1,2,3,\dots\}$, obtain $M_{N}>N$ such that $[p,z_{N}]\cap[ z_{m},z_{n}]=\emptyset$ if $M_{N}\leq m\leq n$. Define $y_{N}=z_{N}\,\hat{\ }\,z_{M_{N}}$ and note that by Lemma \refhat $y_{N}=z_{N}\,\hat{\ }\,z_{m}\,\hat{\ }\,z_{n}=z_{N}\,\hat{\ }\, z_{m}$ if $M_{N}\leq m\leq n$. Note that $ y_{n}\rightarrow y$ and by construction there exists a subsequence $ y_{k_{1}}<y_{k_{2}}\dots$. Let $h:[0,d(x,y))\rightarrow\bigcup_{k=1}^{\infty }[p,y_{n_{k}}]\subset P$ be the natural isometry mapping $ [d(p,y_{k_{n}}),d(p,y_{k_{n+1}})]$ onto $[y_{k_{n}},y_{k_{n+1}}]\subset P$. By construction, $h$ is continuously extendable at $d(p,y)$. \endproof The following lemma establishes that locally interval compact R-trees are open subspaces of their metric completions. \beginlemma \labelnonclosedSuppose that $(P,d,p)$ is a $p$-based incomplete R-tree and $ \partial P=\overline{(P,d,p)}\setminus P$ is not a closed subspace of the metric completion $\overline{(P,d,p)}$. Then $P$ is not locally interval compact. \endlemma \beginproof Obtain $x\in P\cap\overline{\partial P}$. Suppose $\varepsilon>0$. Obtain $ y\in\partial P$ such that $d(x,y)<\varepsilon$. Obtain by Lemma \refdiverge an isometric embedding $ [ 0,d(p,y) ] \rightarrow\overline {P}$ such that $0\mapsto p$, $d(p,y)\mapsto y$, and $[0,d(x,y))$ is order isometric to a closed subspace $\alpha\subset P$. Let $\delta =\varepsilon -d(x,y)$. Obtain $z\in\alpha$ with $d(z,y)<\delta$. Note that if $z<w$ and $ w\in\alpha$, then $ d(w,x)=d(w,z)+d(z,x)<(\varepsilon -d(x,y))+d(x,y)<\varepsilon$. Thus, $[z,y)$ is a closed subspace of $P$, $ [z,y)$ is homeomorphic to $[0,1)$, $[z,y)\subset\overline {B(x,\varepsilon)}$, and $[z,y)$ is not compact. \endproof \beginremark \labelnowayIf $(P,d,p)$ is a $p$-based R tree and $\alpha\subset P$ is homeomorphic to $[0,1)$, then $(\alpha,d)$ is isometric to a unique finite Euclidean half open interval $[0,R)$ for some $R>0$ or the infinite ray $ [0,\infty)$. If $\alpha$ is closed in $P$ and $(\alpha,d)$ is isometric to the finite interval $[0,R)$, then the preimage of the sequence $R-\frac{1}{n}$ shows that $(P,d,p)$ is incomplete. \endremark The following easy lemma is used in the proof of Lemma \refpromote. \beginlemma \labelLcontSuppose that $(X,D)$ is a metric space and $A\subset X$ and $2^{X}$ denotes the collection of compact subsets of $X$ with the Hausdorff distance. Define $L:2^{X}\rightarrow[ 0,\infty)$ as $ L(C)=\inf_{(c,a)\in C\times A}D(c,a)$. Then $L$ is continuous. If $ (P,d,p,\leq,\hat{\ }\,)$ is an R-tree, then $\lambda$ is continuous if $ \lambda:P\rightarrow2^{P}$ is defined as $\lambda(x)=[p,x]$. \endlemma \beginproof By definition the Hausdorff distance $H(C,B)$ [Munkres?] between compacta $\{B,C\}\subset X$ satisfies $0\leq H(B,C)<\varepsilon$ iff for each $b\in B$, there exists $c\in C$ with $D(b,c)<\varepsilon$ and for each $ c\in C$, there exists $b\in B$ with $D(b,c)<\varepsilon$. If $b\in B$ and $ c\in C$ with $D(b,c)<\varepsilon$, then $\llvert L(C)-L(B)\rrvert <\varepsilon$, and in particular $L$ is continuous. If $\{x,y\}\subset P$ with $d(x,y)<\varepsilon$, then $H([p,x],[p,y])=d(x,y)<\varepsilon$, and in particular $\lambda$ is continuous. \endproof The following lemma and its proof also appear in another preprint of the author [Fabel?]. \beginlemma \labelbijectionSuppose that $(P,p,\tau,\leq,\hat{\ }\,)$ is a p-based topological R-tree. Suppose that the continuous function $l:P\rightarrow[ 0,\infty)$ satisfies $x<y\Rightarrow l(x)<l(y)$. Define $d:P\times P\rightarrow[ 0,\infty)$ as $d(x,y)=l(x)+l(y)-2l(x\,\hat{\ }\,y)$. Then $d$ is a metric on the set $P$, inclusion $\kappa:(P,\tau )\rightarrow (P,d)$ is a continuous bijection, each arc $\kappa[ x,y]\subset(P,d)$ is isometric to the Euclidean segment $[0,d(x,y)]$, and $d(x,x\,\hat{\ }\,x_{m})\rightarrow0\Rightarrow x\,\hat{\ }\,x_{m}\rightarrow x$ in $ (P,\tau)$. \endlemma \beginproof Note that $d(x,x)=0$ since $x\,\hat{\ }\,x=x$ and $y\neq x\Rightarrow x\,\hat{\ }\,y<x$ or $x\,\hat{\ }\,y<y$ and hence $d(x,y)>0$. $d(x,y)=d(y,x)$ since $x\,\hat{\ }\,y=y\,\hat{\ }\,x$. Note that $0\leq2(l(y)-l(x\,\hat{\ }\,y))$ since $x\,\hat{\ }\,y\leq y$. Note that $d(x,z)\leq d(x,y)+d(y,z)$ iff $-2l(x\,\hat{\ }\, z)\leq 2l(y)-2l(x\,\hat{\ }\,y)-2l(x\,\hat{\ }\,z)$ iff $0\leq 2(l(y)-l(x \,\hat{\ }\,y))$. The latter holds since $x\,\hat{\ }\,y\leq y$. Thus, $d$ is a metric on the set $P$. If $x_{m}\rightarrow x$ in $(P,\tau)$, then $ x\,\hat{\ }\,x_{m}\rightarrow x $ in $(P,\tau)$. Thus, since $l$ is continuous at $x$, $ l(x)-l(x_{m})\rightarrow0$ and $l(x)-l(x\,\hat{\ }\,x_{m}) \rightarrow0$. Hence, $(l(x)-l(x\,\hat{\ }\,x_{m}))+(l(x_{m})-l(x))+(l(x)-l(x\,\hat{\ }\,x_{m}))=d(x,x_{m})\rightarrow0$. Note that if $\{w,z\}\subset(P,\tau)$ then $w\leq z$ iff $w=z\,\hat{\ }\,w$, and hence by definition, $d(w,z)=l(z)-l(w)$. Thus, if $\{x,y\}\subset (P,\tau)$, then the natural homeomorphism $h_{x,y}:\kappa[ x\,\hat{\ }\, y,x]\rightarrow[ 0,l(x)-l(x\,\hat{\ }\,y)]$ (defined as $ h_{x,y}(z)=l(z)-l(x\,\hat{\ }\,y))$ is an isometry onto the Euclidean segment since $w<u<z\Rightarrow d(z,w)=l(z)-l(w)=(l(z)-l(u))+(l(u)-l(w))=d(z,u)+d(u,z)$. Pasting at $0$ ($ h_{y,x}^{-1}$ union the reverse of $h_{x,y}^{-1}$) yields the natural isometry $[l(x\,\hat{\ }\,y)-l(x),l(y)-l(x\,\hat{\ }\,y)]\rightarrow \kappa [ x,y]$. Suppose $d(x,x\,\hat{\ }\,x_{m})\rightarrow0$. Then $\{x\,\hat{\ }\,x_{m}\}$ is a sequence in the (metrizable) compact arc $[p,x]\subset(P,\tau)$. Since $\kappa$ is continuous at $y$, if $y\in[ p,x]\subset(P,\tau)$ is a subsequential limit of $\{x\,\hat{\ }\,x_{m}\}$, then $y=\kappa (y)=x$. Hence, $x\,\hat{\ }\,x_{m}\rightarrow x$ in $(P,\tau)$. \endproof The standard fact that a space $U$ is topologically complete if $U$ is an open subspace of some complete metric space $(X,d)$ is often established [Munkres?] via a closed embedding $\phi:U\rightarrow X\times R$ with $ u\mapsto(u,\frac{1}{\partial(u,\partial U)})$. For several reasons, this proof does not work “off the shelf” when trying to obtain a complete R-tree metric for a connected open subspace $P\subset Q$ of a complete R-tree $(Q,D)$. Instead, we build a strictly increasing length function $ l:P\rightarrow[ 0,\infty)$ such that $l(x_{n})\rightarrow\infty$ if $ x_{n}\rightarrow\partial P$, apply Lemma \refbijection, and verify completeness of the metric and continuity of the inverse mapping. \beginlemma \labelpromoteSuppose that $(Q,D)$ is a complete metric space, suppose that the subspace $P\subset Q$ is open, nonempty, and dense, and suppose tthat he metric space $(P,D,p,\leq,\hat{\ }\,)$ is a $p$-based R-tree. There exists a topologically compatible metric $d$ on $P$ such that $(P,d,p)$ is a complete R-tree. \endlemma \beginproof Let $\partial P=Q\setminus P$. Define $L:P\rightarrow[ 0,\infty)$ as $L(x)=\inf\{D(y,z)\mid y\in[ p,x]$ and $z\in\partial P\}$. Note that $L>0$ since $[p,x]$ is compact and $\partial P$ is closed. Note that $y\leq x\Rightarrow L(y)\geq L(x)$ since $[p,y]\subset[ p,x]$. Define $ l:P\rightarrow[ 0,\infty)$ as $l(x)=D(p,x)+\frac{1}{L(x)}$. Note that $l$ is continuous since $D$ is continuous and by Remark \refLcont $L$ is continuous. Observe that $\{x,y\}\subset P$ and $x<y\Rightarrow D(p,x)<D(p,y)$ (since $(P,D)$ is an R-tree) and $\frac{1}{L(x)}\leq\frac{1}{L(y)}$ since $ L(y)\geq L(x)$, and hence $l(x)<l(y)$. Thus, applying Lemma \refbijection, the metric $d(x,y)=l(x)+l(y)-2l(x\,\hat{\ }\,y)$ ensures that the inclusion $\kappa :(P,D)\rightarrow(P,d)$ is a continuous bijection, and $\kappa[ x,y]\subset(P,d)$ is isometric to the Euclidean segment $[0,d(x,y)]$. By definition, $D(x,y)=d(x,y)-l(x)-l(y)\leq d(x,y)$. Hence, $\kappa$ is a homeomorphism. Thus, $(P,d)$ is uniquely arcwise connected, and hence $(P,d)$ is an R-tree. Observe that for real numbers, if $0<t<s$, then $1<\frac{1}{t}-\frac {1}{s}$ iff $ st<s-t$. To obtain a contradiction, suppose that $(P,d)$ is incomplete. Let $\overline{(P,d) }$ denote the metric completion of $(P,d)$. By Lemma \refdiverge obtain $ y\in\overline{(P,d)}\setminus P$, and an isometric embedding $ h:[0,d(p,y)]\rightarrow\overline{(P,d)}$, so that $h(0)=p$, $h(d(p,y))=y$ and $h|[0,d(p,y))$ is an order-preserving embedding into $P$. Let $y_{m}=h( \frac{d(p,y)m}{m+1})$. Note that $\{y_{m}\}$ is Cauchy in $(P,d)$ and hence $ \{y_{m}\}$ is Cauchy in $(P,D)$ since $D\leq d$. Note that for all $m\geq1$ and $k\geq1$, $0<L(y_{m})\leq D(y_{m},y_{m+k})$ since $[p,y_{m}]\subset[ p,y_{m+k}]$. Thus, since $\{y_{m}\}$ is Cauchy in $(P,D)$, the sequence $L(y_{m})\rightarrow0$. Hence (applying the continuity of $\times:R\times R\rightarrow R$ and $-:R\times R\rightarrow R $ (familiar multiplication and substraction of real numbers)), for each $ M\geq1$, we obtain $N_{M}>M$ so that $L(y_{M})\times L(y_{n})<L(y_{M})-L(y_{n})$. Thus, if $n\geq N_{M}>M$, then $y_{M}=y_{M}\,\hat{\ }\,y_{n}$, and hence $ d(y_{n},y_{M})=D(y_{n},y_{M})+(\frac{1}{L(y_{n})}-\frac {1}{L(y_{M})})\geq( \frac{1}{L(y_{n})}-\frac{1}{L(y_{M})})>1$, contradicting the fact that $ \{y_{m}\}$ is Cauchy in $(P,d)$. \endproof \sectionProof of Theorem \protect\refmain For $3\Rightarrow2$, suppose that $(P,\tau)$ is a locally interval complete topological R-tree. Obtain by [Oversteegen?] a topologically compatible metric $d$ such that $(P,d)$ is an R-tree. If $(P,d)=\overline{(P,d)}$, then note that $ \overline{(P,d)}$ is open in $\overline{(P,d)}$. If $(P,d)\neq\overline { (P,d)}$, then Lemma \refnonclosed ensures that $P$ is open in $\overline{(P,d)}$. For $2\Rightarrow1$, suppose that $(P,d)$ is an R-tree, open in its metric completion $\overline{(P,d)}$. Apply Lemma \refpromote. For $1\Rightarrow 3$, suppose that $(P,d)$ is a complete R-tree. Note that, by definition, $(P,d)$ is metrizable and uniquely arcwise connected, and $(P,d)$ is locally path connected since open metric balls are path-connected. Recall Remark \ref noway and observe that the bounded open metric balls of radius 1 establish that $(P,d)$ is locally interval compact. \beginthebibliography30
  • A. G. Aksoy and M. A. Khamsi, A selection theorem in metric trees, Proc. Amer. Math. Soc. 134 (2006), no. 10, 2957–2966.
  • J. A. Behrstock, Asymptotic geometry of the mapping class group and Teichmüller space, Geom. Topol. 10 (2006), 1523–1578.
  • V. N. Berestovskii and C. P. Plaut, Covering R-trees, R-free groups, and dendrites, Adv. Math. 224 (2010), no. 5, 1765–1783.
  • M. Bestvina, R-trees in topology, geometry, and group theory, Handbook of geometric topology, pp. 55–91, North-Holland, Amsterdam, 2002.
  • B. H. Bowditch and J. Crisp, Archimedean actions on median pretrees, Math. Proc. Cambridge Philos. Soc. 130 (2001), no. 3, 383–400.
  • M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Springer-Verlag, Berlin, 1999.
  • J. W. Cannon, The theory of negatively curved spaces and groups, Ergodic theory, symbolic dynamics, and hyperbolic spaces, Oxford Sci. Publ., pp. 315–369, Oxford Univ. Press, New York, 1991.
  • I. Chiswell, Introduction to $\Lambda$-trees, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.
  • I. Chiswell, T. W. Müller, and J.-C. Schlage-Puchta, Completeness and compactness criteria for R-trees, preprint.
  • J. J. Dijkstra and K. I. S. Valkenburg, The instability of nonseparable complete Erdős spaces and representations in R-trees, Fund. Math. 207 (2010), no. 3, 197–210.
  • A. Dranishnikov and M. Zarichnyi, Universal spaces for asymptotic dimension, Topology Appl. 140 (2004), no. 2–3, 203–225.
  • C. Drutu and M. V. Sapir, Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups, Adv. Math. 217 (2008), no. 3, 1313–1367.
  • P. Fabel, A short proof characterizing topologically the underlying spaces of R-trees, preprint.
  • H. Fischer and A. Zastrow, Combinatorial R-trees as generalized Cayley graphs for fundamental groups of one-dimensional spaces, Geom. Dedicata 163 (2013), 19–43.
  • M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., 8, pp. 75–263, Springer, New York, 1987.
  • V. Guirardel and A. Ivanov, Non-nesting actions of Polish groups on real trees, J. Pure Appl. Algebra 214 (2010), no. 11, 2074–2077.
  • M. Hamann, On the tree-likeness of hyperbolic spaces, preprint.
  • B. Hughes, Trees and ultrametric spaces: a categorical equivalence, Adv. Math. 189 (2004), no. 1, 148–191.
  • W. Imrich, On metric properties of tree-like spaces, Contributions to graph theory and its applications (Internat. Colloq., Oberhof, 1977), pp. 129–156, Tech. Hochschule Ilmenau, Ilmenau, 1977.
  • I. Kapovich and N. Benakli, Boundaries of hyperbolic groups, Combinatorial and geometric group theory, Contemp. Math., 296, pp. 39–93, Amer. Math. Soc., Providence, RI, 2002.
  • I. Kapovich and M. Lustig, Stabilizers of R-trees with free isometric actions of $F_{N}$, J. Group Theory 14 (2011), no. 5, 673–694.
  • W. A. Kirk, Hyperconvexity of R-trees, Fund. Math. 156 (1998), no. 1, 67–72.
  • –-, Fixed point theorems in $\operatorname{CAT}(0)$ spaces and R-trees, Fixed Point Theory Appl. 4 (2004), 309–316.
  • G. Levitt, Non-nesting actions on real trees, Bull. Lond. Math. Soc. 30 (1998), no. 1, 46–54.
  • J. C. Mayer, L. K. Mohler, L. G. Oversteegen, and E. D. Tymchatyn, Characterization of separable metric R-trees, Proc. Amer. Math. Soc. 115 (1992), no. 1, 257–264.
  • J. C. Mayer, J. Nikiel, and L. G. Oversteegen, Universal spaces for R-trees, Trans. Amer. Math. Soc. 334 (1992), no. 1, 411–432.
  • J. C. Mayer and L. G. Oversteegen, A topological characterization of R-trees, Trans. Amer. Math. Soc. 320 (1990), no. 1, 395–415.
  • J. R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1975.
  • F. Paulin, The Gromov topology on R-trees, Topology Appl. 32 (1989), no. 3, 197–221.
  • K. Ruane, $\operatorname{CAT}(0)$ groups with specified boundary, Algebr. Geom. Topol. 6 (2006), 633–649. \printaddresses