diff --git a/src/autgroups.jl b/src/autgroups.jl
index 0361116..53db353 100644
--- a/src/autgroups.jl
+++ b/src/autgroups.jl
@@ -115,3 +115,61 @@ function evaluate!(
 end
 
 evaluate!(t::NTuple{N, T}, s::GSymbol, A, tmp=one(first(t))) where {N, T} = throw("you need to implement `evaluate!(::$(typeof(t)), ::$(typeof(s)), ::Alphabet, tmp=one(first(t)))`")
+
+# forward evaluate by substitution
+
+struct LettersMap{T, A}
+    indices_map::Dict{Int, T}
+    A::A
+end
+
+function LettersMap(a::FPGroupElement{<:AutomorphismGroup})
+    dom = domain(a)
+    @assert all(isone ∘ length ∘ word, dom)
+    A = alphabet(first(dom))
+    first_letters = first.(word.(dom))
+    img = evaluate!(dom, a)
+
+    # (dom[i] → img[i] is a map from domain to images)
+    # we need a map from alphabet indices → (gens, gens⁻¹) → images
+    # here we do it for elements of the domai
+    # (trusting it's a set of generators that define a)
+    @assert length(dom) == length(img)
+
+    indices_map = Dict(A[A[fl]] => word(im) for (fl, im) in zip(first_letters, img))
+    # inverses of generators are dealt lazily in getindex
+
+    return LettersMap(indices_map, A)
+end
+
+
+function Base.getindex(lm::LettersMap, i::Integer)
+    # here i is an index of an alphabet
+    @boundscheck 1 ≤ i ≤ length(KnuthBendix.letters(lm.A))
+
+    if !haskey(lm.indices_map, i)
+        img = if haskey(lm.indices_map, inv(lm.A, i))
+            inv(lm.A, lm.indices_map[inv(lm.A, i)])
+        else
+            @warn "LetterMap: neither $i nor its inverse has assigned value"
+            one(valtype(lm.indices_map))
+        end
+        lm.indices_map[i] = img
+    end
+    return lm.indices_map[i]
+end
+
+function (a::FPGroupElement{<:AutomorphismGroup})(g::FPGroupElement)
+    @assert object(parent(a)) === parent(g)
+    img_w = evaluate(word(g), LettersMap(a))
+    return parent(g)(img_w)
+end
+
+evaluate(w::AbstractWord, lm::LettersMap) = evaluate!(one(w), w, lm)
+
+function evaluate!(res::AbstractWord, w::AbstractWord, lm::LettersMap)
+    for i in w
+        append!(res, lm[i])
+    end
+    return res
+end
diff --git a/test/AutFn.jl b/test/AutFn.jl
index a33dc65..a269163 100644
--- a/test/AutFn.jl
+++ b/test/AutFn.jl
@@ -144,6 +144,36 @@
         @test all(g->isone(g*inv(g)), B_2)
     end
 
+    @testset "Forward evaluate" begin
+        N = 3
+        F = FreeGroup(N)
+        G = SpecialAutomorphismGroup(F)
+
+        a = gens(G, 1) # ϱ₁₂
+
+        f = gens(F)
+
+        @test a(f[1]) == f[1]*f[2]
+        @test all(a(f[i]) == f[i] for i in 2:length(f))
+
+        S = let s = gens(G)
+            [s; inv.(s)]
+        end
+
+        @test all(
+            map(first(Groups.wlmetric_ball(S, radius=2))) do g
+                lm = Groups.LettersMap(g)
+                img = evaluate(g)
+
+                fimg = [F(lm[first(word(s))]) for s in gens(F)]
+
+                succeeded = all(img .== fimg)
+                @assert succeeded "forward evaluation of $(word(g)) failed: \n img=$img\n fimg=$(tuple(fimg...))"
+                succeeded
+            end
+        )
+    end
+
     @testset "GroupsCore conformance" begin
         test_Group_interface(A)
         g = A(rand(1:length(alphabet(A)), 10))
@@ -153,37 +183,3 @@
     end
 
 end
-
-# using Random
-# using GroupsCore
-#
-# A = New.SpecialAutomorphismGroup(FreeGroup(4), maxrules=2000, ordering=KnuthBendix.RecursivePathOrder)
-#
-# # for seed in 1:1000
-# let seed = 68
-#     N = 14
-#     Random.seed!(seed)
-#     g = A(rand(1:length(KnuthBendix.alphabet(A)), N))
-#     h = A(rand(1:length(KnuthBendix.alphabet(A)), N))
-#     @info "seed=$seed" g h
-#     @time isone(g*inv(g))
-#     @time isone(inv(g)*g)
-#     @info "" length(word(New.normalform!(g*inv(g)))) length(word(New.normalform!(inv(g)*g)))
-#     a = commutator(g, h, g)
-#     b = conj(inv(g), h) * conj(conj(g, h), g)
-#
-#     @info length(word(a))
-#     @info length(word(b))
-#
-#     w = a*inv(b)
-#     @info length(word(w))
-#     New.normalform!(w)
-#     @info length(word(w))
-#
-#
-#     #
-#     # @time ima = evaluate(a)
-#     # @time imb = evaluate(b)
-#     # @info "" a b ima imb
-#     # @time a == b
-# end