Real tree

<h2 id="definition-and-examples">Definition and examples</h2>
<h3>Formal definition</h3>

<p>A metric space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a real tree if it is a <a href="/facts/Geodesic_metric_space/e0aAePjd">geodesic space</a> where every triangle is a tripod. That is, for every three points 
  
    
      
        x
        ,
        y
        ,
        ρ
        ∈
        X
      
    
    {\displaystyle x,y,\rho \in X}
  
 there exists a point 
  
    
      
        c
        =
        x
        ∧
        y
      
    
    {\displaystyle c=x\wedge y}
  
 such that the geodesic segments 
  
    
      
        [
        ρ
        ,
        x
        ]
        ,
        [
        ρ
        ,
        y
        ]
      
    
    {\displaystyle [\rho ,x],[\rho ,y]}
  
 intersect in the segment 
  
    
      
        [
        ρ
        ,
        c
        ]
      
    
    {\displaystyle [\rho ,c]}
  
 and also 
  
    
      
        c
        ∈
        [
        x
        ,
        y
        ]
      
    
    {\displaystyle c\in [x,y]}
  
. This definition is equivalent to 
  
    
      
        X
      
    
    {\displaystyle X}
  
 being a "zero-hyperbolic space" in the sense of Gromov (all triangles are "zero-thin"). 
Real trees can also be characterised by a <a href="/facts/Topology/2EqW47cU">topological</a> property. A metric space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a real tree if for any pair of points 
  
    
      
        x
        ,
        y
        ∈
        X
      
    
    {\displaystyle x,y\in X}
  
 all <a href="/facts/Topological_embedding/HKeVKc42">topological embeddings</a> 
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
 of the segment 
  
    
      
        [
        0
        ,
        1
        ]
      
    
    {\displaystyle [0,1]}
  
 into 
  
    
      
        X
      
    
    {\displaystyle X}
  
 such that 
  
    
      
        σ
        (
        0
        )
        =
        x
        ,
        
        σ
        (
        1
        )
        =
        y
      
    
    {\displaystyle \sigma (0)=x,\,\sigma (1)=y}
  
 have the same image (which is then a geodesic segment from 
  
    
      
        x
      
    
    {\displaystyle x}
  
 to 
  
    
      
        y
      
    
    {\displaystyle y}
  
).
</p>
<h3>Simple examples</h3>
<ul><li>If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a connected graph with the combinatorial metric then it is a real tree if and only if it is a tree (i.e. it has no <a href="/facts/Cycle_(graph_theory)/Is44edrv">cycles</a>). Such a tree is often called a simplicial tree. They are characterised by the following topological property: a real tree 
  
    
      
        T
      
    
    {\displaystyle T}
  
 is simplicial if and only if the set of singular points of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 (points whose complement in 
  
    
      
        X
      
    
    {\displaystyle X}
  
 has three or more connected components) is closed and discrete in 
  
    
      
        X
      
    
    {\displaystyle X}
  
.</li>
<li>The 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
-tree obtained in the following way is nonsimplicial. Start with the interval [0, 2] and glue, for each positive integer <i>n</i>, an interval of length 1/<i>n</i> to the point 1 − 1/<i>n</i> in the original interval. The set of singular points is discrete, but fails to be closed since 1 is an ordinary point in this 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
-tree. Gluing an interval to 1 would result in a <a href="/facts/Closed_set/upYnRTUj">closed set</a> of singular points at the expense of discreteness.</li>
<li>The <a href="/facts/Paris_metric/cu2tXXwA">Paris metric</a> makes the plane into a real tree. It is defined as follows: one fixes an origin 
  
    
      
        P
      
    
    {\displaystyle P}
  
, and if two points are on the same ray from 
  
    
      
        P
      
    
    {\displaystyle P}
  
, their distance is defined as the Euclidean distance. Otherwise, their distance is defined to be the sum of the Euclidean distances of these two points to the origin 
  
    
      
        P
      
    
    {\displaystyle P}
  
.</li>
<li>The plane under the Paris metric is an example of a <a href="/facts/Hedgehog_space/cu2tXXwA">hedgehog space</a>, a collection of line segments joined at a common endpoint. Any such space is a real tree.</li></ul>
<h2 id="characterizations">Characterizations</h2>

<p>Here are equivalent characterizations of real trees which can be used as definitions:
</p><p>1) <i>(similar to <a href="/facts/Tree_(data_structure)/RcXz9noF">trees</a> as graphs)</i> A real tree is a <a href="/facts/Intrinsic_metric/V0TslFne">geodesic</a> <a href="/facts/Metric_space/gnBBRXtc">metric space</a> which contains no subset <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphic</a> to a circle.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p><p>2) A real tree is a connected metric space 
  
    
      
        (
        X
        ,
        d
        )
      
    
    {\displaystyle (X,d)}
  
 which has the four points condition<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> (see figure): 
</p>
For all 
  
    
      
        x
        ,
        y
        ,
        z
        ,
        t
        ∈
        X
        ,
      
    
    {\displaystyle x,y,z,t\in X,}
  
  
  
    
      
        d
        (
        x
        ,
        y
        )
        +
        d
        (
        z
        ,
        t
        )
        ≤
        max
        [
        d
        (
        x
        ,
        z
        )
        +
        d
        (
        y
        ,
        t
        )
        
        ;
        
        d
        (
        x
        ,
        t
        )
        +
        d
        (
        y
        ,
        z
        )
        ]
      
    
    {\displaystyle d(x,y)+d(z,t)\leq \max[d(x,z)+d(y,t)\,;\,d(x,t)+d(y,z)]}
  
.
<p>3) A real tree is a connected <a href="/facts/%25CE%2594-hyperbolic_space/Daqo6Ah2">0-hyperbolic</a> metric space<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> (see figure). Formally, 
</p>
For all 
  
    
      
        x
        ,
        y
        ,
        z
        ,
        t
        ∈
        X
        ,
      
    
    {\displaystyle x,y,z,t\in X,}
  
  
  
    
      
        (
        x
        ,
        y
        
          )
          
            t
          
        
        ≥
        min
        [
        (
        x
        ,
        z
        
          )
          
            t
          
        
        
        ;
        
        (
        y
        ,
        z
        
          )
          
            t
          
        
        ]
        ,
      
    
    {\displaystyle (x,y)_{t}\geq \min[(x,z)_{t}\,;\,(y,z)_{t}],}

<p>where 
  
    
      
        (
        x
        ,
        y
        
          )
          
            t
          
        
      
    
    {\displaystyle (x,y)_{t}}
  
 denotes the <a href="/facts/Gromov_product/KKvRID5K">Gromov product</a> of 
  
    
      
        x
      
    
    {\displaystyle x}
  
 and 
  
    
      
        y
      
    
    {\displaystyle y}
  
 with respect to 
  
    
      
        t
      
    
    {\displaystyle t}
  
, that is, 
  
    
      
        
          
            
              1
              2
            
          
          
            (
            
              d
              (
              x
              ,
              t
              )
              +
              d
              (
              y
              ,
              t
              )
              −
              d
              (
              x
              ,
              y
              )
            
            )
          
          .
        
      
    
    {\displaystyle \textstyle {\frac {1}{2}}\left(d(x,t)+d(y,t)-d(x,y)\right).}

</p><p>4) <i>(similar to the characterization of <a href="/facts/Tree_(graph_theory)/TCWk4Joy">plane trees</a> by their contour process).</i> Consider a positive excursion of a function. In other words, let 
  
    
      
        e
      
    
    {\displaystyle e}
  
 be a continuous real-valued function and 
  
    
      
        [
        a
        ,
        b
        ]
      
    
    {\displaystyle [a,b]}
  
 an interval such that 
  
    
      
        e
        (
        a
        )
        =
        e
        (
        b
        )
        =
        0
      
    
    {\displaystyle e(a)=e(b)=0}
  
 and 
  
    
      
        e
        (
        t
        )
        >
        0
      
    
    {\displaystyle e(t)>0}
  
 for 
  
    
      
        t
        ∈
        ]
        a
        ,
        b
        [
      
    
    {\displaystyle t\in ]a,b[}
  
.
</p><p>For 
  
    
      
        x
        ,
        y
        ∈
        [
        a
        ,
        b
        ]
      
    
    {\displaystyle x,y\in [a,b]}
  
, 
  
    
      
        x
        ≤
        y
      
    
    {\displaystyle x\leq y}
  
, define a <a href="/facts/Metric_space/gnBBRXtc">pseudometric</a> and an <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> with: 
</p>

d
          
            e
          
        
        (
        x
        ,
        y
        )
        :=
        e
        (
        x
        )
        +
        e
        (
        y
        )
        −
        2
        min
        (
        e
        (
        z
        )
        
        ;
        z
        ∈
        [
        x
        ,
        y
        ]
        )
        ,
      
    
    {\displaystyle d_{e}(x,y):=e(x)+e(y)-2\min(e(z)\,;z\in [x,y]),}

x
        
          ∼
          
            e
          
        
        y
        ⇔
        
          d
          
            e
          
        
        (
        x
        ,
        y
        )
        =
        0.
      
    
    {\displaystyle x\sim _{e}y\Leftrightarrow d_{e}(x,y)=0.}

<p>Then, the <a href="/facts/Quotient_space_(topology)/muE0qQVC">quotient space</a> 
  
    
      
        (
        [
        a
        ,
        b
        ]
        
          /
        
        
          ∼
          
            e
          
        
        
        ,
        
        
          d
          
            e
          
        
        )
      
    
    {\displaystyle ([a,b]/\sim _{e}\,,\,d_{e})}
  
 is a real tree.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>   Intuitively, the <a href="/facts/Local_minima/ADlgnyzV">local minima</a> of the excursion <i>e</i> are the parents of the <a href="/facts/Local_maxima/ADlgnyzV">local maxima</a>. Another visual way to construct the real tree from an excursion is to "put glue" under the curve of <i>e</i>, and "bend" this curve, identifying the glued points (see animation).
</p>

<h2 id="examples">Examples</h2>
<p>Real trees often appear, in various situations, as limits of more classical metric spaces.
</p>
<h3>Brownian trees</h3>
<p>A <a href="/facts/Brownian_tree/5b1CAhKG">Brownian tree</a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> is a random metric space whose value is a (non-simplicial) real tree almost surely. Brownian trees arise as limits of various random processes on finite trees.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>
<h3>Ultralimits of metric spaces</h3>
<p>Any <a href="/facts/Ultralimit/aC31JMos">ultralimit</a> of a sequence 
  
    
      
        (
        
          X
          
            i
          
        
        )
      
    
    {\displaystyle (X_{i})}
  
 of 
  
    
      
        
          δ
          
            i
          
        
      
    
    {\displaystyle \delta _{i}}
  
-<a href="/facts/Hyperbolic_metric_space/Daqo6Ah2">hyperbolic</a> spaces with 
  
    
      
        
          δ
          
            i
          
        
        →
        0
      
    
    {\displaystyle \delta _{i}\to 0}
  
 is a real tree. In particular, the <a href="/facts/Ultralimit/aC31JMos">asymptotic cone</a> of any hyperbolic space is a real tree.
</p>
<h3>Limit of group actions</h3>
<p>Let 
  
    
      
        G
      
    
    {\displaystyle G}
  
 be a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a>. For a sequence of based 
  
    
      
        G
      
    
    {\displaystyle G}
  
-spaces 
  
    
      
        (
        
          X
          
            i
          
        
        ,
        
          ∗
          
            i
          
        
        ,
        
          ρ
          
            i
          
        
        )
      
    
    {\displaystyle (X_{i},*_{i},\rho _{i})}
  
 there is a notion of convergence to a based 
  
    
      
        G
      
    
    {\displaystyle G}
  
-space 
  
    
      
        (
        
          X
          
            ∞
          
        
        ,
        
          x
          
            ∞
          
        
        ,
        
          ρ
          
            ∞
          
        
        )
      
    
    {\displaystyle (X_{\infty },x_{\infty },\rho _{\infty })}
  
 due to M. Bestvina and F. Paulin. When the spaces are hyperbolic and the actions are unbounded the limit (if it exists) is a real tree.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
</p><p>A simple example is obtained by taking 
  
    
      
        G
        =
        
          π
          
            1
          
        
        (
        S
        )
      
    
    {\displaystyle G=\pi _{1}(S)}
  
 where 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is a <a href="/facts/Compact_space/d0cgXJH7">compact</a> surface, and 
  
    
      
        
          X
          
            i
          
        
      
    
    {\displaystyle X_{i}}
  
 the universal cover of 
  
    
      
        S
      
    
    {\displaystyle S}
  
 with the metric 
  
    
      
        i
        ρ
      
    
    {\displaystyle i\rho }
  
 (where 
  
    
      
        ρ
      
    
    {\displaystyle \rho }
  
 is a fixed hyperbolic metric on 
  
    
      
        S
      
    
    {\displaystyle S}
  
).
</p><p>This is useful to produce actions of hyperbolic groups on real trees. Such actions are analyzed using the so-called <a href="/facts/Rips_machine/FeUJLVlo">Rips machine</a>. A case of particular interest is the study of degeneration of groups acting <a href="/facts/Group_action/Vh9VtUUw">properly discontinuously</a> on a <a href="/facts/Hyperbolic_space/kI7tnwdk">real hyperbolic space</a> (this predates Rips', Bestvina's and Paulin's work and is due to J. Morgan and <a href="/facts/Peter_Shalen/zHvUVWXm">P. Shalen</a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>).
</p>
<h3>Algebraic groups</h3>
<p>If 
  
    
      
        F
      
    
    {\displaystyle F}
  
 is a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> with an <a href="/facts/Ultrametric_space/QhJQBG9N">ultrametric</a> <a href="/facts/Valuation_(algebra)/FXadrz5c">valuation</a> then the <a href="/facts/Building_(mathematics)/916RYbEN">Bruhat–Tits building</a> of 
  
    
      
        
          
            S
            L
          
          
            2
          
        
        (
        F
        )
      
    
    {\displaystyle \mathrm {SL} _{2}(F)}
  
 is a real tree. It is simplicial if and only if the valuations is discrete.
</p>
<h2 id="generalisations">Generalisations</h2>
<h3>Λ-trees</h3>
<p>If 
  
    
      
        Λ
      
    
    {\displaystyle \Lambda }
  
 is a <a href="/facts/Totally_ordered_abelian_group/APXGv4UI">totally ordered abelian group</a> there is a natural notion of a distance with values in 
  
    
      
        Λ
      
    
    {\displaystyle \Lambda }
  
 (classical metric spaces correspond to 
  
    
      
        Λ
        =
        
          R
        
      
    
    {\displaystyle \Lambda =\mathbb {R} }
  
). There is a notion of 
  
    
      
        Λ
      
    
    {\displaystyle \Lambda }
  
-tree<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> which recovers simplicial trees when 
  
    
      
        Λ
        =
        
          Z
        
      
    
    {\displaystyle \Lambda =\mathbb {Z} }
  
 and real trees when 
  
    
      
        Λ
        =
        
          R
        
      
    
    {\displaystyle \Lambda =\mathbb {R} }
  
. The structure of <a href="/facts/Finitely_presented_group/f9emPUoI">finitely presented groups</a> acting <a href="/facts/Group_action/Vh9VtUUw">freely</a> on 
  
    
      
        Λ
      
    
    {\displaystyle \Lambda }
  
-trees was described. <a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> In particular, such a group acts freely on some  
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
-tree.
</p>
<h3>Real buildings</h3>
<p>The axioms for a <a href="/facts/Building_(mathematics)/916RYbEN">building</a> can be generalized to give a definition of a real building. These arise for example as asymptotic cones of higher-rank <a href="/facts/Symmetric_spaces/AWYuHfNp">symmetric spaces</a> or as Bruhat-Tits buildings of higher-rank groups over valued fields.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Dendroid_(topology)/EJjpp1Yd">Dendroid (topology)</a></li>
<li><a href="/facts/Tree-graded_space/3g2Vk9N3">Tree-graded space</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Chiswell, Ian (2001). Introduction to [lambda]-trees. Singapore: World Scientific. ISBN 978-981-281-053-3. OCLC 268962256. <a href="978-981-281-053-3" target="_blank">978-981-281-053-3</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Peter Buneman, A Note on the Metric Properties of Trees, Journal of combinatorial theory, B (17), p. 48-50, 1974. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Evans, Stevan N. (2005). Probability and Real Trees. École d’Eté de Probabilités de Saint-Flour XXXV. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Evans, Stevan N. (2005). Probability and Real Trees. École d’Eté de Probabilités de Saint-Flour XXXV. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Aldous, D. (1991), "The continuum random tree I", Annals of Probability, 19: 1–28, doi:10.1214/aop/1176990534 <a href="/wiki/David_Aldous" target="_blank">/wiki/David_Aldous</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Aldous, D. (1991), "The continuum random tree III", Annals of Probability, 21: 248–289 <a href="/wiki/David_Aldous" target="_blank">/wiki/David_Aldous</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Bestvina, Mladen (2002), "
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
-trees in topology, geometry and group theory", Handbook of Geometric Topology, Elsevier, pp. 55–91, ISBN 9780080532851 <a href="9780080532851" target="_blank">9780080532851</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Shalen, Peter B. (1987), "Dendrology of groups: an introduction", in Gersten, S. M. (ed.), Essays in Group Theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer-Verlag, pp. 265–319, ISBN 978-0-387-96618-2, MR 0919830 <a href="978-0-387-96618-2" target="_blank">978-0-387-96618-2</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Chiswell, Ian (2001), Introduction to Λ-trees, River Edge, NJ: World Scientific Publishing Co. Inc., ISBN 981-02-4386-3, MR 1851337 <a href="981-02-4386-3" target="_blank">981-02-4386-3</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>O. Kharlampovich, A. Myasnikov, D. Serbin, Actions, length functions and non-archimedean words IJAC 23, No. 2, 2013.{{citation}}:  CS1 maint: multiple names: authors list (link) <a href="/wiki/Template:Citation" target="_blank">/wiki/Template:Citation</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
</ol>

Real tree open-in-new

Real tree