80

vertex, 34

morphism

of graphs, 6

covering, 6

immersion, 6

inert immersion, 21

locally injective,

6

locally surjective, 6

of groupoids, 7

of G-sets, 5

naturally equivalent,

abstract maps of graphs, 10

groupoid morphisms, 7

Neumann,

H., 73

Neumann,

W.D., 2,

19

Nielsen-Schreier Theorem,

17

non-negative matrix, 27

non-trivial, element of groupoid, 7

normal form, 9

normalized,

Perron-Frobenius eigenvector, 29

pseudometric, 31

number of Z-occurrences, 9

occur,

edge, 9

turn, 10

on (vertex on turn), 10

open,

graph element, 5

groupoid element, 7

orbit, 5

oriented, (3,

24

outer group automorphism, 34

partial order on

Q(/),

34

permutation matrix, 27

Perron-Frobenius,

eigenvector, 29

normalized, 29

eigenvalue, 29

of ((3, Z, Zo),

33

pseudometric, 33

Theorem, 28

positive matrix, 27

predivision, 13

in

Q(/),

38

take place in

Zo,

38

pre-order on

Q(/),

34

presentation, 5

pretrivial

for a groupoid morphism, 7

for an abstract map of graphs, 10

proper

segment, 10

subgraph, 6

pseudometric, 31

Pullback Theorem, 19

INDEX

pullback

for graphs, ii, 1, 19

for groupoids, 19

quotient,

graph, 16

set, 5

rank,

of a free group, 1

of free groupoid, 15

(see also

reduced rank)

re-orienting, 11

reduced graph rank, 15

reduced rank,

of free group, 1

of free groupoid, 15

of graph, 15

ofF-set, 15

reducible matrix, 27

Reidemeister class,

24

Reidemeister's Theorem, 16

relative train track, 46

remove valence two,

42

either way,

42

represent, 11

Scott conjecture, 1

Scott, G.P., 1

segment, 1210

initial, 10

terminal, 10

self-map (of a graph), 11

inner, 11

similar, 11

set with a group action, 5

associated to,

groupoid morphism,

9

abstract map of graphs, 25 21

inert, 19

reduced rank of, 15

Shalen, P.B., 1

similar,

groupoid endomorphisms, 8

self-maps, 11

smooth,

element, 47

turn, 47

Solitar,

D.,

2, 22

stabilizer, 5

Stallings,

J.R.,

ii, 1, 2, 13, 19, '72

stratum, 46

subcategory, 10

subdivision, 13

at a non-vertex fixed-point,

37

in

Q(¢),

36

predivision, 13

in

Q(/),

38

take place in

Zo,

38