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
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