Cutting up

• The geometrical problem
• Cutting up the fundamental domain
• The picture in the upper half-space model of
• The case of no circles
• The case of one circle
• The case of two tangent circles
• The case of two non-tangent circles
• The case of three or more circles
• The obvious strategy doesn't work
• Keeping the tubes more or less vertical
• Working our way down
• Associating pieces of the plane to relevant hemispheres
• Taking care of the babies
• Room to grow
• Annexing annuli
• Avoiding conflicts between neighboring hemispheres
• Overlap of renounced territory
• Avoiding overlap
• An elementary geometric lemma
• How little area to ask for
• Laying out the tubes
• Taking stock
• What is the constant?