# The symmetry of intersection numbers in group theory

@article{Scott1998TheSO, title={The symmetry of intersection numbers in group theory}, author={Peter Scott}, journal={Geometry \& Topology}, year={1998}, volume={2}, pages={11-29} }

For suitable subgroups of a nitely generated group, we dene the intersection number of one subgroup with another subgroup and show that this number is symmetric. We also give an interpretation of this number.

## 21 Citations

Splittings of groups and intersection numbers

- Mathematics
- 2000

We prove algebraic analogues of the facts that a curve on a surface with selfintersection number zero is homotopic to a cover of a simple curve, and that two simple curves on a surface with…

GROWTH OF INTERSECTION NUMBERS FOR FREE GROUP AUTOMORPHISMS

- Mathematics
- 2010

For a fully irreducible automorphism φ of the free group Fk we compute the asymptotics of the intersection number n �→ i(T, Tφ n ) for the trees T and T in Outer space. We also obtain qualitative…

The quasi-isometry invariance of commensurizer subgroups

- Mathematics
- 2010

We prove that commensurizers of two-ended subgroups with at least three coends in one-ended, finitely presented groups are invariant under quasi-isometries. We discuss a variety of applications of…

Splittings and C-complexes

- Mathematics
- 2009

The intersection pattern of the translates of the limit set of a quasiconvex subgroup of a hyperbolic group can be coded in a natural incidence graph, which suggests connections with the splittings…

Canonical splittings of groups and 3-manifolds

- Mathematics
- 2001

We introduce the notion of a `canonical' splitting over Z or ZxZ for a finitely generated group G. We show that when G happens to be the fundamental group of an orientable Haken manifold M with…

Splittings of non-finitely generated groups

- Mathematics
- 2012

In geometric group theory one uses group actions on spaces to gain information about groups. One natural space to use is the Cayley graph of a group. The Cayley graph arguments that one encounters…

J un 2 00 9 Splittings and C-Complexes

- 2009

The intersection pattern of the translates of the limit set of a quasi-convex subgroup of a hyperbolic group can be coded in a natural incidence graph, which suggests connections with the splittings…

Scott and Swarup's regular neighbourhood as a tree of cylinders

- Mathematics
- 2008

Let G be a finitely presented group. Scott and Swarup have constructed a canonical splitting of G which encloses all almost invariant sets over virtually polycyclic subgroups of a given length. We…

0 Splittings of Groups and Intersection Numbers

- 2008

We prove algebraic analogues of the facts that a curve on a surface with self-intersection number zero is homotopic to a cover of a simple curve, and that two simple curves on a surface with…

## References

SHOWING 1-10 OF 28 REFERENCES

Cyclic Splittings of Finitely Presented Groups and the Canonical JSJ-Decomposition

- Mathematics
- 1997

The classification of stable actions of finitely presented groups on ℝ-trees has found a number of applications. Perhaps one of the most striking of these applications is the theory of canonical…

Groups of Cohomological Dimension One

- Mathematics
- 1972

Cohomology theory.- Ends.- The structure theorem.- The augmentation ideal.- The finitely generated case.- The countable case.- Splitting theorems.- The main theorems.

On the ends of pairs of groups

- Mathematics
- 1993

Abstract We develop a technique for calculating the ends of a pair of groups (G,H) for special types of (G, H).

An algebraic annulus theorem

- Mathematics
- 1996

We present an extension of Dunwoody's theory of tracks and use it to prove an analogue of the annulus theorem for hyperbolic groups.

CLOSED GEODESICS ON SURFACES

- Mathematics
- 1982

1—> M representing a. is either an embedding or a double cover of a one-sided embedded curve K. In the second case, C bounds a Moebius band in M and K is isotopic to the centre of this band.

JSJ-splittings for finitely presented groups over slender groups

- Mathematics
- 1999

Abstract. We generalize the JSJ-splitting of Rips and Sela to give decompositions of finitely presented groups which capture splittings over certain classes of small subgroups. Such classes include…

Splittings of Poincaré Duality Groups II

- Mathematics
- 1988

Soit (G,S) une PD u .2− 1 -paire et T un PD n−1 -sous-groupe de G tel que T∼S pour tout ∈S. Alors G admet une decomposition, adaptee a SS, sur un sous-groupe commensurable avec T si et seulement si…

Ends of locally compact groups and their coset spaces

- Mathematics
- 1974

Freudenthal [5, 7] defined a compactification of a rim-compact space, that is, a space having a base of open sets with compact boundary. The additional points are called ends and Freudenthal showed…

Homological Group Theory: Topological methods in group theory

- Mathematics
- 1979

List of contributors 1. Left relatively convex subgroups Yago Antolin, Warren Dicks and Zoran Sunic 2. Groups with context-free co-word problem and embeddings into Thompson's group V Rose…

Least area incompressible surfaces in 3-manifolds

- Mathematics
- 1983

Let M be a Riemannian manifold and let F be a closed surface. A map f: F---,M is called least area if the area of f is less than the area of any homotopic map from F to M. Note that least area maps…