Groups of proper homotopy equivalences of graphs and Nielsen Realization

Abstract

For a locally finite connected graph $X$ we consider the group $Maps(X)$ of proper homotopy equivalences of $X$. We show that it has a natural Polish group topology, and we propose these groups as an analog of big mapping class groups. We prove the Nielsen Realization theorem: if $H$ is a compact subgroup of $Maps(X)$ then $X$ is proper homotopy equivalent to a graph $Y$ so that $H$ is realized by simplicial isomorphisms of $Y$.

Figures (6)

• Figure 1: A graph with a finitely generated fundamental group. In this case $\partial X_g = \emptyset$ and the space of ends can be any closed subset of the Cantor set.
• Figure 2: This is a core graph, $\partial X= \partial X_g$. By deleting a part of the Cantor tree we get a standard model for $(A,A)$ where $A$ is any closed subspace of the Cantor set.
• Figure 3: In this case both $\partial X_g$ and its complement $DX:=\partial X\smallsetminus \partial X_g$ are non-empty. By deleting loops/subtrees from the space in Figure \ref{['fig:minipage2']} we get a model for any pair of closed subsets $\partial X = A \supset B = \partial X_g$ of the Cantor set.
• Figure 4: The homotopy $H$. The brackets signify that we take the immersed path homotopic to the given one rel endpoints.
• Figure 5: $H:I\times I\to X$ denotes the homotopy from $q\nu$ to $h'hq\nu$. The path denoted by $\beta$ is mapped by $H$ to a path homotopic to $q\nu$. Since $H$ maps vertical segments $\{t\}\times I$ to paths whose images have $D$-length less than $3$, the dotted subpaths are mapped by $H$ to paths with $D$-length smaller than $3$. The dashed part of $\beta$ is null homotopic. Since $q\nu$ is immersed, its $D$-length is $<6$.

Theorems & Definitions (92)

• Definition 1.1
• Definition 2.1: Characteristic pairs
• Theorem 2.2
• Corollary 2.3
• Definition 2.4
• Definition 2.5: Standard Models
• Corollary 2.6
• Proposition 2.7
• proof
• Lemma 2.8