Lvzhou Chen

Stable commutator length

in graphs of groups

NCNGT 1

Lvzhou Chen

NCNGT

May 27, 2020

Lvzhou Chen NCNGT 2

Stable commutator length (scl)

Definition

Admissible surfaces2

2

1

1

Multiplicative!

Lvzhou Chen NCNGT 3

Stable commutator length (scl)

Definition

Example

Challenge: prove lower bounds of scl or compute it.

Admissible surfaces

Space of surfaces has little structure.

Multiplicative!

Lvzhou Chen NCNGT 4

Properties

Lvzhou Chen NCNGT 5

Gromov–Thurston norm

Remove “embedded” by Gabai–Thurston

Gromov Norm

Example

extremal

Lvzhou Chen NCNGT 6

Extremal surfaces

Definition

Note: extremalfinite cover

extremal

Proof: Compression+LERF

Proposition

Lvzhou Chen NCNGT 7

Computations and rationality

Theorem (Calegari’08)

Gromov’s Question

Application (surface subgroups)

Lvzhou Chen NCNGT 8

Main theorem

Theorem (C.’19)

Examples

f.i

Lvzhou Chen NCNGT 9

Proof ideas

Lvzhou Chen NCNGT 10

Relative admissible surfaces

Lvzhou Chen NCNGT 11

Simple normal form

Lvzhou Chen NCNGT 12

Simple normal form

Lvzhou Chen NCNGT 13

Linear programming

Lvzhou Chen NCNGT 14

Dimension reduction (BS case)

Lvzhou Chen NCNGT 15

Asymptotic promotion

Lvzhou Chen NCNGT 16

Asymptotic promotion

Lvzhou Chen NCNGT 17

Asymptotic promotion

Lvzhou Chen NCNGT 18

Asymptotic promotion

Lvzhou Chen NCNGT 19

Asymptotic promotion