Free factor complex

In mathematics, the free factor complex (sometimes also called the complex of free factors) is a free group counterpart of the notion of the curve complex of a finite type surface. The free factor complex was originally introduced in a 1998 paper of Allen Hatcher and Karen Vogtmann. Like the curve complex, the free factor complex is known to be Gromov-hyperbolic. The free factor complex plays a significant role in the study of large-scale geometry of \operatorname{Out}(F_n) .

In mathematics, the free factor complex (sometimes also called the complex of free factors) is a free group counterpart of the notion of the curve complex of a finite type surface. The free factor complex was originally introduced in a 1998 paper of Allen Hatcher and Karen Vogtmann. Like the curve complex, the free factor complex is known to be Gromov-hyperbolic. The free factor complex plays a significant role in the study of large-scale geometry of \operatorname{Out}(F_n) .

Formal definition

For a free group G a proper free factor of G is a subgroup A\le G such that A\ne \{1\}, A\ne G and that there exists a subgroup B\le G such that G=A\ast B . Let n\ge 3 be an integer and let F_n be the free group of rank n . The free factor complex \mathcal F_n for F_n is a simplicial complex where: (1) The 0-cells are the conjugacy classes in F_n of proper free factors of F_n , that is \mathcal F_n^{(0)}=\{[A] | A\le F_n \text{ is a proper free factor of } F_n \}. (2) For k\ge 1 , a k -simplex in \mathcal F_n is a collection of k+1 distinct 0-cells \{v_0, v_1, \dots, v_k\}\subset \mathcal F_n^{(0)} such that there exist free factors A_0,A_1,\dots, A_k of F_n such that v_i=A_i for i=0,1,\dots, k , and that A_0\le A_1\le \dots \le A_k . [The assumption that these 0-cells are distinct implies that A_i\ne A_{i+1} for i=0,1,\dots, k-1 ]. In particular, a 1-cell is a collection \{[A], [B]\} of two distinct 0-cells where A,B\le F_n are proper free factors of F_n such that A\lneq B . For n=2 the above definition produces a complex with no k -cells of dimension k\ge 1 . Therefore, \mathcal F_2 is defined slightly differently. One still defines \mathcal F_2^{(0)} to be the set of conjugacy classes of proper free factors of F_2 ; (such free factors are necessarily infinite cyclic). Two distinct 0-simplices \{v_0,v_1\}\subset \mathcal F_2^{(0)} determine a 1-simplex in \mathcal F_2 if and only if there exists a free basis a,b of F_2 such that v_0=[\langle a\rangle], v_1=[\langle b\rangle] . The complex \mathcal F_2 has no k -cells of dimension k\ge 2 . For n\ge 2 the 1-skeleton \mathcal F_n^{(1)} is called the free factor graph for F_n .

Main properties

For every integer n\ge 3 the complex \mathcal F_n is connected, locally infinite, and has dimension n-2 . The complex \mathcal F_2 is connected, locally infinite, and has dimension 1. For n=2 , the graph \mathcal F_2 is isomorphic to the Farey graph. There is a natural action of \operatorname{Out}(F_n) on \mathcal F_n by simplicial automorphisms. For a k-simplex \Delta=\{[A_0],\dots, [A_k]\} and \varphi\in \operatorname{Out}(F_n) one has \varphi \Delta:=\{[\varphi(A_0)],\dots, [\varphi(A_k)]\} . For n\ge 3 the complex \mathcal F_n has the homotopy type of a wedge of spheres of dimension n-2 . For every integer n\ge 2 , the free factor graph \mathcal F_n^{(1)} , equipped with the simplicial metric (where every edge has length 1), is a connected graph of infinite diameter. For every integer n\ge 2 , the free factor graph \mathcal F_n^{(1)} , equipped with the simplicial metric, is Gromov-hyperbolic. This result was originally established by Mladen Bestvina and Mark Feighn; see also for subsequent alternative proofs. An element \varphi\in \operatorname{Out}(F_n) acts as a loxodromic isometry of \mathcal F_n^{(1)} if and only if \varphi is fully irreducible. There exists a coarsely Lipschitz coarsely \operatorname{Out}(F_n) -equivariant coarsely surjective map \mathcal{FS}_n\to \mathcal F_n^{(1)} , where \mathcal{FS}_n is the free splittings complex. However, this map is not a quasi-isometry. The free splitting complex is also known to be Gromov-hyperbolic, as was proved by Handel and Mosher. Similarly, there exists a natural coarsely Lipschitz coarsely \operatorname{Out}(F_n) -equivariant coarsely surjective map CV_n\to \mathcal F_n^{(1)} , where CV_n is the (volume-ones normalized) Culler–Vogtmann Outer space, equipped with the symmetric Lipschitz metric. The map \pi takes a geodesic path in CV_n to a path in \mathcal FF_n contained in a uniform Hausdorff neighborhood of the geodesic with the same endpoints. The hyperbolic boundary \partial \mathcal F_n^{(1)} of the free factor graph can be identified with the set of equivalence classes of "arational" F_n -trees in the boundary \partial CV_n of the Outer space CV_n . The free factor complex is a key tool in studying the behavior of random walks on \operatorname{Out}(F_n) and in identifying the Poisson boundary of \operatorname{Out}(F_n) .

Other models

There are several other models which produce graphs coarsely \operatorname{Out}(F_n) -equivariantly quasi-isometric to \mathcal F_n^{(1)} . These models include: The graph whose vertex set is \mathcal F_n^{0} and where two distinct vertices v_0,v_1 are adjacent if and only if there exists a free product decomposition F_n=A\ast B\ast C such that v_0=[A] and v_1=[B] . The free bases graph whose vertex set is the set of F_n -conjugacy classes of free bases of F_n , and where two vertices v_0,v_1 are adjacent if and only if there exist free bases \mathcal A, \mathcal B of F_n such that v_0=[\mathcal A], v_1=[\mathcal B] and \mathcal A\cap \mathcal B\ne \varnothing .

References

See also

Mapping class group

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