Cut points in Bowditch Boundary of Relatively hyperbolic groups 2

Understanding Swenson (large excerpt.. some diagrams .. some remarks). Please be cautious. Potentially wrong remarks lie ahead.

Continuum

A continuum is a compact connected Hausdorff space.

Cut Point

In a continuum Z, \( c \in Z \) is a cut point if \( Z = A \cup B \) where A and B are non-singleton continua and \( A \cap B = \{c\} \). If in addition \( D \subset A – \{c\} \) and \( E \subset B – \{ c \} \), we say that c separates D from E.

For the remainder of this section, Z will be a metric continuum, G will be a group (possibly trivial) of homeomorphisms of Z, and \( C \subset Z \) will be a G-equivariant (GC = C) set of cut points of Z.

Is a member of (interval)

For \( a, b \in Z \) and \( c \in C \), we define \( c \in (a, b) \) if there exist non-singleton continua A containing a and B containing b with \( A \cup B = Z \) and \( A \cap B = \{ c \} \).

Any point in an interval is a cut point by definition of interval.

We define the closed and half open intervals in the obvious way i.e., \( [a, b] = \{a, b \} \cup (a, b) \), and \( [a, b) = \{a\} \cup (a, b) \) for \( a \neq b \) ( \([a, a) = \phi \) )

Notice that if \( c \in (a, b) \) then for any subcontinuum \( Y \subset Z \) from a to b (\(a, b \in Y\) ), \( c \in Y \). Why? Suppose Y is a subcontinuum containing a and b. \( c \in (a, b) \) implies there are continua A and B such that \( a \in A, b \in B, A \cup B = Z, A \cap B = \{ c \} \).

\( A \cap Y \) and \(B \cap Y \) compact and Hausdorff (hence are subcontinua) containing a and b respectively. Their union is Y (as all y in Y are either in A or B as A union B is Z and Y is contained in Z). Their intersection is either \( \phi \) or {c}.

confusion

Equivalence of points

For \( a, b \in Z – C \) we say that a is equivalent to b, a ~ b, if \( (a, b) = \phi \). That is there is no cut point between a and b.For \( c \in C \), c is equivalent only to itself. This is clearly an equivalence relation, so let P bet the set of equivalence classes of Z. We will abuse notation and say \( C \subset P \) since each element of C is its own equivalence class.

Observe that for \( a, b, d\in Z \) if a ~ b, then (a, d) = (b, d). We can therefore translate the interval relation on Z to P and we also enlarge it as follows.

Membership in P (interval in P)

For \( x, y, z \in P \), we say \( y \in (x, z) \) if either

1. \( y \in C\) where \( y \in (a, b) \) for some \( a, b \in Z \) with \( a \in x \) and \( b \in z \) (note that x, and z are equivalence classes ) or
2. \( y \notin C and if \( a, b, d \in Z \) with \( a\in x , b \in Y \) and \( d \in z \), then \( [a, b) \cup (b, d] = \phi \)

Since C was chosen to be G invariant, the action of G on Z gives an action of G on P which preserves the interval structure ( we have not given P a topology so it doesn’t make sense to ask if the action is by homeomorphism).